4d fuel optimal trajectory generation from waypoint ... - ubi · v abstract the purpose of this...

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UNIVERSIDADE DA BEIRA INTERIOR

Engenharia

4D Fuel Optimal Trajectory Generation from

Waypoint Networks

Kawser Ahmed

Dissertação para obtenção do grau de mestre em

Engenharia Aeronáutica (Ciclo de estudos integrado)

Orientador: Prof. Doutor Kouamana Bousson

Covilhã, fevereiro de 2016

http://www.ubi.pt/en/Courses#M__Integrado

iii

Acknowledgements

I would like to express my sincere appreciation to Professor Kouamana Bousson for his support and

guidance throughout this thesis and throughout my education at The University of Beira Interior.

Finally, I would like to thank my family and friends for their support during my education at The

University of Beira Interior.

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Abstract

The purpose of this thesis is to develop a trajectory optimization algorithm that finds a fuel optimal

trajectory from 4D waypoint networks, where the arrival time is specified for each waypoint in the

network. Generating optimal aircraft trajectory that minimizes fuel burn and associated

environmental emissions helps the aviation industry cope with increasing fuel costs and reduce

aviation induced climate change, as CO2 is directly related to the amount of fuel burned, therefore

reduction in fuel burn implies a reduction in CO2 emissions as well.

A single source shortest path algorithm is presented to generate the optimal aircraft

trajectory that minimizes the total fuel burn between the initial and final waypoint in pre-defined

4D waypoint networks. In this work the 4D waypoint networks only consist of waypoints for climb,

cruise and descent phases of the flight without the takeoff and landing approach. The fuel optimal

trajectory is generated for three different lengths of flights (short, medium and long haul flight) for

two different commercial aircraft considering no wind.

The Results about the presented applications show that by flying a fuel optimal trajectory,

which was found by implying a single source shortest path algorithm (Dijkstra’s algorithm) can lead

to reduction of average fuel burn of international flights by 2.8% of the total trip fuel. By using the

same algorithm in 4D waypoints networks it is also possible to generate an optimal trajectory that

minimizes the flight time. By flying this trajectory average of 2.6% of total travel time can be saved,

depends on the trip length and aircraft types.

Keywords

Fuel Conservation; Cost Index; Dijkstra’s algorithm; 4D Waypoint navigation; Base of Aircraft Data

(BADA);

vii

Resumo

Esta tese tem como objetivo desenvolver um algoritmo de otimização de trajetória que permita

encontrar uma trajetória de combustível ótima em uma redes de waypoint em 4D, onde o tempo de

chegada é específico para cada waypoint da rede. Ao criar uma trajetória ótima que minimize o

consumo de combustível da aeronare e as suas respetivas emissões poluentes, ajuda a indústria da

aviação não só a lidar com o aumento nos custos dos combustíveis, bem como a reduzir a sua

contribuição nas alterações climáticas, pois o CO2 está diretamente relacionado com a quantidade

de combustível queimado, logo uma redução no seu consumo implica que haja também uma redução

nas emissões de CO2.

O algoritmo “single source shortest path” é utilizado de forma a gerar uma trajetória ótima,

que minimize o consumo de combustível entre o waypoint inicial e final de redes pré-definida de

waypoint em 4D. Neste trabalho, esta redes consiste num conjunto de waypoints inseridos apenas

nas fases de voo de subida, cruzeiro e descida, ignorando assim as fases de descolagem e aterragem.

A trajetória de combustível ótima é criada para dois aviões comerciais diferentes em três distâncias

de voo também diferentes (voo curto, médio e longo), sem considerar o vento.

Os resultados deste trabalho mostram que ao voar numa trajetória de combustível ótima,

obtida através do algoritmo “single source shortest path” (Dijkstra’s algorithm), é possível reduzir o

consumo total de combustível numa média de 2.8%, em voos internacionais. Utilizando o mesmo

algoritmo numa rede de waypoints em 4D é também possível encontrar uma trajetória ótima que

minimize o tempo de voo numa media de 2.6% do tempo total, consoante a distância da viagem e do

tipo de aeronave.

Palavras-chave

Conservação de combustível; Cost Index; Dijkstra’s algorithm; Navegação por waypoints em 4D; Base

of Aircraft Data (BADA);

ix

Index

List of Figures……………………………………………………………………………………………………………………………………..xi

List of Tables………………………………………………………………………………………………………………………………………xiii

List of Acronyms ………………………………………………………………………………………………………………………………..xv

List of Symbols……………………………………………………………………………………………………………………………………xvii

1 Introduction ...................................................................................................... 1

1.1 Motivation ................................................................................................. 1

1.2 Fuel Saving in Different Phases of Commercial Flight ............................................. 3

1.2.1 Cruise Phase ............................................................................................ 4

1.2.2 Takeoff and Climb Phase ............................................................................ 6

1.2.3 Descent and Approach Phase ........................................................................ 7

1.3 Trajectory Optimization ................................................................................ 9

1.4 Objectives ................................................................................................ 10

1.5 Outline .................................................................................................... 11

2 Dijkstra’s Algorithm ........................................................................................... 13

2.1 Pseudo-code of Dijkstra’s Algorithm ................................................................. 14

2.2 Representation of Graph ............................................................................... 15

2.3 Implementation of Dijkstra’s in Flight Trajectory Optimization ................................ 16

3 Modeling of the 4D Waypoints Network .................................................................... 17

3.1 Navigation Model ........................................................................................ 18

3.2 Flight Constraints ....................................................................................... 18

3.3 Arrival Time of Each Waypoint ....................................................................... 19

3.4 Engine Thrust ............................................................................................ 21

3.4.1 Maximum Climb and Take-off Thrust ............................................................. 21

3.4.2 Maximum Cruise Thrust ............................................................................. 22

3.4.3 Descent Thrust ........................................................................................ 22

3.5 Fuel Consumption Model ............................................................................... 23

3.5.1 Thrust Specific Fuel Consumption ................................................................. 23

3.5.2 Nominal Fuel Flow Rate ............................................................................. 24

3.6 Consumed Fuel Between Waypoints ................................................................. 25

4 Optimal Trajectory Generation ............................................................................. 27

4.1 Selection of Flights for Analysis ...................................................................... 27

4.2 Selection of Aircraft for Analysis ..................................................................... 28

4.3 Fuel Optimal Trajectory Generation ................................................................. 29

4.4 Time Optimal Trajectory Generation ................................................................ 29

4.5 Minimization of Delay of each Waypoint in Fuel and Time Optimal Trajectories. .......... 30

x

4.5.1 Determination of coefficients...................................................................... 30

5 Simulation and Result ......................................................................................... 33

5.1 Short Haul Flight ........................................................................................ 33

5.1.1 Fuel Optimal Trajectory ............................................................................ 35

5.1.2 Time Optimal Trajectory ........................................................................... 36

5.2 Medium Haul Flight ..................................................................................... 37



5.3 Long Haul Flight ......................................................................................... 40



6 Conclusion and Discussions ................................................................................... 45

6.1 Future work .............................................................................................. 46

7 References ...................................................................................................... 47

xi

List of Figures

Figure 1-1: Fuel Prices over the years in Dollar ................................................................. 1

Figure 1-2: Effect of the winglet on the vortex ................................................................. 2

Figure 1-3: Different phases of commercial flights from takeoff to landing. ............................. 4

Figure 1-4: Comparison of different cruise speeds ............................................................. 4

Figure 1-5: Optimum altitude determination at constant Mach number ................................... 5

Figure 1-6: Descent profile at given IAS .......................................................................... 7

Figure 1-7: Typical descent phase of commercial flight ...................................................... 8

Figure 2-1: Pseudo-code for Dijkstra’s algorithm .............................................................. 14

Figure 2-2: The execution of Dijkstra's algorithm ............................................................. 15

Figure 2-3: Representation of graph G into matrix ............................................................ 16

Figure 5-1: 3D fuel optimal trajectory in geocentric coordinates for short haul flight. ................ 35

Figure 5-2: 3D time optimal trajectory in geocentric coordinates for short haul flight. ............... 36

Figure 5-3: 3D fuel optimal trajectory in geocentric coordinates for medium haul flight. ............ 39

Figure 5-4: 3D time optimal trajectory in geocentric coordinates for medium haul flight. ........... 40

Figure 5-5: 3D fuel optimal trajectory in geocentric coordinates for long haul flight. ................. 42

Figure 5-6: 3D fuel optimal trajectory in geocentric coordinates for long haul flight. ................. 43

xiii

List of Tables

Table 1-1: ECON cruise Mach in different cruise wind conditions ........................................... 5

Table 1-2: Impact of takeoff flap settings on fuel burn ....................................................... 6

Table 1-3: Fuel saving potential of two climb profile ......................................................... 7

Table 1-4: Fuel savings estimates for delayed flaps approach procedure ................................. 9

Table 4-1: Characteristics of airplane A1 and airplane A2 ................................................... 28

Table 5-1: List of waypoints in 1st trajectory of short haul flight .......................................... 34

Table 5-2: List of waypoints in 2nd trajectory of short haul flight ......................................... 34

Table 5-3: Total fuel consumed in different trajectories for short haul flight. .......................... 35

Table 5-4: Total time needed in different trajectories for short haul flight. ............................ 36

Table 5-5: List of waypoints in 1st trajectory of medium haul flight ...................................... 37

Table 5-6:List of waypoints in 2nd trajectory of medium haul flight ...................................... 38

Table 5-7: Total Fuel consumed in different trajectories for medium haul flight. ...................... 38

Table 5-8: Total time needed in different trajectories for medium haul flight. ........................ 39

Table 5-9: List of waypoints in 1st trajectory of long haul flight ........................................... 41

Table 5-10: List of waypoints in 2nd trajectory of long haul flight ........................................ 41

Table 5-11: Total Fuel consumed in different trajectories for long haul flight. ......................... 42

Table 5-12: Total time needed in different trajectories for long haul flight. ............................ 43

xv

List of Acronyms

AGL Above Ground Level

ATC Air Traffic control

BADA Base of Aircraft Data

CAEP Committee on Aviation Environment Protection

CAS Calibrated Air Speed

CDU Control Display Unit

CI Cost Index

FMC Flight Management Computer

GHG Green House Gas

GPS Global Positioning System

GTF Geared Turbofan Technique

IAS Indicated Air Speed

ICAO International Civil Aviation Organization

IPCC Intergovernmental Panel on Climate Change

LRC Long Range Cruise

MRC Maximum Range Cruise

SID Standard Instrument Departure

SR Specific Range

STAR Standard Terminal Arrival Route

TAS True Air Speed

TOD Top of Descent

TPBVPs Two-Point Boundary Value Problems

xvii

List of Symbols

Arrival time tolerance interval Latitude [deg]

Flight path angle [deg]

Thrust specific fuel consumption [kg/(min.KN)]

Longitude [deg]

Air density [kg/m3]

Arrival time at waypoint [min]

Heading [deg]

d Distance between waypoints [nm]

T Temperature deviation from standard atmosphere [K]

a Acceleration [m/s]

a Earth semi major axis [nm]

LC Lift coefficient

1fC ,

2fC Thrust specific fuel consumption coefficient

3fC ,

4fC Descent fuel flow coefficient

crfC Cruise fuel flow correction coefficient

,1cTC ,,2cTC ,

,3cTC Climb thrust coefficient

,4cTC ,,5cTC Thrust temperature coefficient

rcTC Maximum cruise thrust coefficient

.des appTC Approach thrust coefficient

. highdesTC High altitude descent thrust coefficient

. lddesTC Landing thrust coefficient

. lowdesTC Low altitude descent thrust coefficient

D Drag

df Consumed fuel between waypoints [kg]

d Travel time between waypoints [min]

E Edges between vertices

e Eccentricity

/ap ldf Approach and landing fuel flow rate [kg/min]

crf Cruise fuel flow rate [kg/min]

minf Minimum fuel flow rate [kg/min]

nomf Nominal fuel flow rate [kg/min]

G Graph

h altitude [feet]

xviii

pH Geo-potential pressure altitude [feet]

,p desH Transition altitude [feet]

dM Descent Mach number [Mach]

P Waypoint

eR Earth radius [nm]

S Wing area [m2]

s Source vertex

Thr Thrust [KN]

( )cruiseThr MAX Maximum cruise thrust [KN]

,appdesThr Approach thrust [KN]

,des highThr High altitude descent thrust [KN]

,lddesThr Landing thrust [KN]

,lowdesThr Low altitude descent thrust [KN]

maxclimbThr Maximum climb thrust [KN]

maxclimb( )Thr ISA Maximum climb thrust at standard atmospheric condition [KN]

V Flight velocity [Knots]

V Vertices of the graph

CASV Calibrated air speed [Knots]

TASV True air speed [Knots]

W Aircraft Nominal weight [kg]

w Weight of the graph

, ,X Y Z Geocentric Coordinates [nm]

Chapter 1: Introduction

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Chapter 1

1 Introduction

1.1 Motivation

Fuel saving on flight mission for commercial aircraft becomes an important factor nowadays in

aviation mainly because of two reason, one is ever increasing fuel prices and other is to reduce the

emission rates of greenhouse gases (GHG) into the atmosphere.

Improving aircraft operational efficiency has become a dominant theme in air

transportation, as the airlines around the world have seen the price of fuel has risen sharply during

the last decades (figure 1-1). The fuel cost represents around 30% of the operating costs for the

airlines, thus the airlines are looking for different ways to reduce the flight operating costs by

reducing the fuel consumption during the flight, also mounting scientific evidence of global climate

change has spurred increased awareness of the importance of manmade greenhouse gas (GHG)

emissions such as CO2, resulting in significant pressure to reduce emissions. International civil aviation

organization (ICAO) and committee on aviation Environmental protection (CAEP’s) estimated that

currently, aviation accounts for about 2% of total global CO2 emissions and about 12% of the CO2 from

all transportation source [1], and by 2050, aviation’s contribution could increase to 5% of the total

human generated global warming. In terms of climate change, the Intergovernmental Panel on

climate change (IPCC) estimated an increase in the earth’s temperature of approximately 1.6 degrees

Fahrenheit by 2050, of which about 0.09 degrees would be attributed to aviation. This increased fuel

Figure 1-1: Fuel Prices over the years in Dollar


2

prices and environmental concerns have pushed airlines to reduce fuel consumption and to find

margins for performance improvements [2].

Technological improvement such as development of more efficient engines, lighter

materials, new aerodynamic designs, and the optimization of the flight trajectories can lead to

reduction of fuel consumption. The engine builders propose new engine Pratt&Whitney, which

developed the geared turbofan technique (GTF) which is endowed with speed reducer between the

fan and the low pressure compressor, each component running at its optimal speed and improve the

reactor performance, which results in reduction of fuel consumption. Currently Southwest airlines

uses this engine for Airplane A1 series aircraft in order to increase engine efficiency and save fuel

[3]. New aerodynamic design, such as winglets which reduce the aircraft drag by altering the airflow

near the wingtip and decreases the vortex which makes it possible to reduce the fuel consumption

(figure 1-2). The reduction of the fuel capacity at takeoff and weight reduction also helps to reduce

the fuel consumption. Cathay Pacific airlines strip paint off some aircraft to reduce fuel burn. The

polished silver fuselage makes the Boeing 747 about 200 kg lighter and saves more than HK$ 1.5

million on its annual fuel bill [4].

Efforts to modernize the aircraft fleet are limited by extremely slow and expensive process of

new aircraft adoption, which can take decades, therefore it is important to find different alternatives

to reduce the fuel consumption in current aircraft, which will likely to share the sky with most modern

aircraft in near future. One of these alternatives is to optimize flight trajectories and traffic control

procedure. Therefore, flight trajectory optimization with emphasis on fuel become really important

to reduce the fuel consumption. The existing flight planning techniques are suboptimal. Hence, an

fuel optimal flight path can significantly save fuel.

Figure 1-2: Effect of the winglet on the vortex


3

1.2 Fuel Saving in Different Phases of Commercial Flight

In commercial flight the rate of fuel burn mainly depends on ambient temperature, aircraft speed,

and aircraft altitude. It also depend on aircraft weight which changes as fuel burned. The wind may

provide a head or tail wind component which in turn will increase or decrease the fuel consumption

by increasing or decreasing the air distance to be flown.

A practical solution that reduces the cost associated with time and fuel consumption during

flight is the cost index (CI). The value of the CI reflects the relative effects of fuel cost on overall

trip cost as compared to time-related direct operating cost. The cost index (CI) is shown in (equation

1.1).

~ (€ / )

~ (€ / )

TimeCost hrCI

FuelCost kg

(1.1)

The flight crew enters the company calculated CI into the control display unit (CDU) of the

flight management computer (FMC). The FMC uses this number and other performance parameters

to calculate economy climb, cruise and descent speeds.

For all the aircraft models, the minimum value of cost index equal to zero results in maximum

range airspeed and minimum trip fuel, but this configuration ignores the time cost. In the case when

the cost index is maximum, the flight time is minimum, the velocity and the Mach number are

maximum, but ignores the fuel cost.

The time related cost in CI depend on various things, such as flight crew wages, that can have

an hourly cost associated with them, engines, auxiliary power units, and the airplanes can be leased

by the hour and the maintenance can be accounted by the hour. As a result each of these items may

have a high direct time costs, in such case CI is large to minimize the time. In the case where most

costs are fixed, the CI is very low to minimize the fuel cost. The cost index allows finding a

compromise between the fuel burn and the time according to the costs of both, to reduce the flight

cost [5].

Commercial flights follow the following phases: Take-off, climb, cruise, descent and approach.

Each of these phases may be divided into several flight segments. Mathematically each flight segment

can be described by two constant control variables selected from among engine thrust settings, Mach

number or calibrated airspeed, and altitude rate or flight path angle. Different phases of flight are

shown in (figure 1-3). The following subsections provide more details about fuel saving on different

phases of flight.


4

1.2.1 Cruise Phase

In general except for short flight trajectory, the largest percentage of the trip time and the trip fuel

are consumed typically in cruise phase of flight, hence it is really important to have the best cruise

condition to reduce the fuel consumption. The two variables which affect the travel time and the

fuel burn most, are the cruise speed, and the altitude or flight level. The correct selection of the

cruise parameters is therefore fundamental in minimizing fuel or operating cost, study shows that

aircraft consume less fuel when flown slower or when flown higher. However there are limits to these

laws. Flying slower than the maximum range speed will increase the block fuel, as will flying higher

than an optimum altitude.

There are two theoretical speed selections for cruise phase of flight. The traditional speed is

long range cruise (LRC), which is the speed that will provide the furthest distance traveled for a given

amount of fuel burned and maximum range cruise (MRC) is the minimum fuel burned for a given

cruise distance. Since fuel is not the only direct cost associated with a flight, a further refinement in

the speed for most economical operation is ECON speed, based on entered CI. Which include some

tradeoffs between trip time and trip fuel. The LRC speed is almost universally higher than the speed

that will result from using CI selected by most carriers. But the MRC speed has the better fuel mileage

Figure 1-3: Different phases of commercial flights from takeoff to landing.

Figure 1-4: Comparison of different cruise speeds


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for all the cruise speed (figure 1-4). So the best strategy to conserve fuel is to select the MRC speed,

which has CI value equal to zero.

The specific range (SR) changes with the altitude at a constant Mach number, it is apparent

that, for each weight, there is an altitude where SR is maximum. This altitude is referred to as “

optimum altitude” (figure 1-5). The optimum altitude is not constant and changes over the period of

a long flight as atmospheric condition and the weight of the aircraft changes. A large change in

temperature will significantly alter the optimum altitude with a decrease in temperature

corresponding to an increase in altitude.

When the aircraft flies at the optimum altitude, it is operated at the maximum lift to drag

ratio corresponding to the selected Mach number. The maximum range cruise (MRC) Mach number

gives the best specific range. Nevertheless, for practical operation, a long range cruise (LRC) Mach

number procedure gains a significant increase in speed compare to MRC with only a 1% loss in specific

range. Like the MRC, the LRC speed also decreases with decreasing weight, at constant altitude [6]

The LRC and the MRC speed calculated by the FMC is typically not adjusted for winds at cruise

altitude. They are ideal only for Zero wind conditions. While the ECON speed is optimized for all

cruise wind conditions (table 1-1). For example, in the presence of strong tailwind, the ECON speed

will be reduced in order to maximize the advantage gained from the tailwind during cruise.

Conversely, the ECON speed will be increased when flying into a head wind in cruise to minimize the

penalty associated with the head wind [7].

Table 1-1: ECON cruise Mach in different cruise wind conditions

Cost Index 100 Kt Tailwind Zero wind 100 Kt Headwind

0 0.773 0.773 0.785

80 0.787 0.796 0.803

MAX 0.811 0.811 0.811

Figure 1-5: Optimum altitude determination at constant Mach number


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1.2.2 Takeoff and Climb Phase

A standard instrument departure (SID) procedure or a departure procedure which defines a pathway

from an airport runway to a waypoint on an airways, so that an aircraft can join the airway system

in a controlled manner.

The climb phase has some restrictions. Upper and lower bound of flight path angle is

primarily maintained because of passenger comfort and are also in effect throughout an entire flight.

Another restriction imposed on the entire flight limits calibrated airspeed ( CASV ) 250 knots or less

below altitude of 10,000 feet (3,048 m).

An important consideration when seeking fuel saving in the takeoff and climb phase of flight

is the takeoff flap settings. The lower the flap setting, the lower the drag, resulting less fuel burned.

(table 1-2) Shows the effect of takeoff flap setting on fuel burn from break release to a pressure

altitude of 10,000 feet (3,048 m), assuming an acceleration altitude of 3,000 feet (914 m) above

ground level (AGL).

Table 1-2: Impact of takeoff flap settings on fuel burn

Airplane Model Takeoff Flap

settings

Takeoff Gross

weight (kg)

Fuel used (kg) Fuel Differential

(kg)

737-800 Winglets

5

72,575

578 -

10 586 8

15 588 10

747-400 10 328,855 2,555 -

20 2,618 63

Another area in the takeoff and climb phase where fuel burn can be reduced is in the climb

out and cleanup operation. If the flight crew performs acceleration and flap retraction at a lower

altitude than the typical 3,000 feet (914 m), the fuel burn is reduced because of the drag is being

reduced earlier in the climb out phase.

(table 1-3) Shows two standard climb profile for both airplanes. Profile 1 is a climb profile

with acceleration and flap retraction beginning at 3,000 feet (914 m) above ground level (AGL),

Profile 2 is a climb profile with acceleration to flap retraction speed beginning at 1,000 feet (305

m) AGL. Generally when airplanes fly profile 2 they use 3 to 4 percent less fuel than profile 1 [8].


7

Table 1-3: Fuel saving potential of two climb profile

Airplane model Takeoff Gross

weight (kg)

Profile

type

Takeoff Flap

setting

Fuel used

(kg)

Fuel

Differential (kg)

737-800 Winglets 72,575

1

10

2,374 -

2 2,307 67

747-400 328,855

1

10

9,549 -

2 9,313 236

1.2.3 Descent and Approach Phase

There are two main parameters to act on when willing to lower the fuel burn for the descent phase:

the speed and the descent gradient whose combination determines the thrust required. The normal

procedure for a descent is to select 3o descent slope and to maintain the Indicated air speed (IAS) by

adjusting the thrust.

Descending at a higher slope enables to save fuel, as less thrust is required for the descent.

The top of descent (TOD) occurs later and the flight at cruise level is longer. Beside, at a given

gradient of descent, the slower the IAS selected, the less fuel is burnt during the descent, as less

thrust is required (figure 1-6).

Most of the airlines follow standard terminal arrival route (STAR) or an arrival procedure

which defines a pathway from a waypoint on an airway to an airport runway, so that aircraft can

leave the airway system in a controlled manner. STAR usually covers the phase of the flight that lies

between top of descent from cruise and the final approach to a runway for landing.

Figure 1-6: Descent profile at given IAS


8

The entire descent is considered to use idle thrust, but in practice a varying throttle will

generally be required. As the cruise phase ends, the aircraft enters a constant altitude deceleration

until it reaches its specified descent Mach number dM , which is less than the cruise Mach. This

transitions into a constant Mach descent segment and is followed by a constant Calibrated Airspeed

CASV segment as the aircraft descends. At approximately 10,000 feet the aircraft enters a constant

altitude or shallow descent segment until it decelerates to a Calibrated Airspeed CASV of 250 knots

as is required under 10,000 feet. The aircraft then maintains a Calibrated Airspeed CASV of under

250 knots until it descends to approximately 3,000 feet where it begins the final landing Approach

(figure 1-7).

Low-drag delayed flaps or noise abatement approach is another type of descent approach.

This approach is flown in a low drag configuration at a speed considerably higher than the final

approach speed. At the appropriate time, power is reduced to idle and the flaps and gear are

extended while decelerating to final approach speed. The throttles are partially advanced to initiate

engine acceleration prior to selecting final approach flaps and are further advanced to normal

approach power as the final approach speed is reached. The configuration and power changes are

scheduled so as to stabilize in the landing configuration at a target altitude above 152 m (500 feet),

selected by the pilot. The remainder of the approach is conventional [9].

Depending on the flap settings and airplane model, the delayed flaps approach uses 7 to 173

fewer kilograms of fuel than the standard approach with the same flap settings (table 1-3). Flight

crews can vary their approach procedures and flap selections to match the flight’s strategic

Figure 1-7: Typical descent phase of commercial flight


9

objectives, which almost always include fuel conservation, noise abatement, and emissions reduction

[10].

Table 1-4: Fuel savings estimates for delayed flaps approach procedure

Airplane Engine Landing weight

(kg)

Landing Flap (deg)

Procedure Fuel Burned (kg)

Fuel differential

(kg)

737-800 CFM56-7B24 54,431

30 Standard 104 - Delayed 97 7


747-400 CF6-80C2B1F 204,116



1.3 Trajectory Optimization

Classical aircraft trajectory optimizations are solved by applying calculus of variations to determine

the optimality conditions, requiring the solution of non-linear two-point boundary value problems

(TPBVPs) [11]. Alternatively, a more general solution to aircraft trajectory optimization can be

obtained by singular perturbation theory which approximates solutions of high order problems by the

solution of a series of lower order systems with the system dynamics separated into low and fast

modes [12], [13].

With the tremendous advancement in numerical computing power, TPBVPs can be converted

to nonlinear programming problems that are solvable even for problems with many variables and

constraints using numerical algorithms such as direct collocation methods. Neglecting aircraft

dynamics and applying shortest path algorithms in graph theory, an optimal trajectory can be

approximated by the path that minimizes the total link cost connecting the origin and destination in

a pre-defined network. The graph methods often require large computation time and memory space

but guarantee global optimal solutions. In this thesis the single source shortest path algorithm was

used to generate the fuel optimal trajectory.

This study is restricted to the climb, cruise and descent phases of the flight and ignores the

takeoff and landing approach, and assuming the initial and final waypoints are at altitude of 3000

feet, where in the initial waypoint the aircraft begins the climb phase and in the final waypoint the

aircraft begins the landing approach.


10

1.4 Objectives

The objective of this thesis is to find an optimal trajectory in 4D waypoint networks by using a single

source shortest path algorithm (Dijkstra’s algorithm) which minimizes the fuel burn. The objective

can be fulfilled by achieving the following three goals:

1. To select a network of waypoints in 3D ( , , )T

k k k kP h between initial and final waypoints,

then calculating the associated arrival time k in each waypoint.

2. To establish a method to calculate the associated consumed fuel kdf between the waypoints

in the network using Base of Aircraft Data (BADA) .

3. Implementation of the Dijkstra’s shortest path algorithm to find out the fuel optimal

trajectory in 4D waypoint network between the initial and final waypoint.

This work primarily attempts to quantify benefits of fuel optimal trajectory which was found by

implying the Dijkstra’s shortest path algorithm. In this work, a benefit is meant to imply a reduction

in fuel burn due to using the Dijkstra’s shortest path algorithm to the actual unimproved flight.

As CO2 is directly related to the amount of fuel burned, reduction in fuel consumption implies a

reduction in CO2 emissions as well. Therefore this analysis answers the question: How much can fuel

burn and CO2 emission be reduced in flight if aircraft are operated in fuel optimal trajectory? This

work also attempts to quantify the benefits of time optimal trajectory which reduces the total travel

time of the trajectory.

Optimal trajectories were generated for three different lengths of flight, they are short haul

flight (Lisbon – Geneva), medium haul flight (Lisbon – Stockholm) and long haul flight (Lisbon –

Montreal). In this work the trajectories are in climb, cruise and descent phases of the flight,

considering no wind.

The key aspect of this thesis is a detailed comparison between actual flight trajectories and

corresponding more efficient trajectories, thus giving the most realistic estimate of improvement

potential.


11

1.5 Outline

The thesis is organized as follows, with major contribution of each chapter highlighted:

Chapter 1 describes the motivation of the work and the state of art about fuel saving in flight

trajectory.

Chapter 2 described the Dijkstra’s shortest path algorithm and convert a waypoint network

graph into a matrix.

Chapter 3 briefly describes the modeling of waypoint network by calculating the associated

travel time and consumed fuel between the waypoints.

Chapter 4 describes the fuel and time optimal trajectory generation by using Dijkstra’s

algorithm and minimization of delay of each waypoint in these trajectories.

Chapter 5 presents the simulation results for the fuel and time optimal trajectory and

compare the results with different trajectories.

Chapter 6 gives a summary of the work, provides conclusions, discussions, and present a

future work recommendations.


12

Chapter 2: Dijkstra’s Algorithm

13

Chapter 2

2 Dijkstra’s Algorithm

Dijkstra’s algorithm is a simple greedy algorithm to solve single source shortest paths problem, it was

conceived by a Dutch computer scientist Edsger Dijkstra in 1956 and was first published in 1959. The

algorithm exists in many variants; Dijkstra's original variant finds the shortest path between two

vertices, but a more common variant fixes a single vertex as the "source" vertex and finds shortest

paths from the source to all other vertices in the graph, producing a shortest-path tree.

Dijkstra’s algorithm uses the greedy approach to solve the single source shortest path

problem. It can also be used for finding costs of shortest paths from a single source vertex to a single

destination vertex by stopping the algorithm once the shortest path to the destination vertex has

been determined. For example, if the vertices of the graph represent cities and edge path costs

represent driving distances between pairs of cities connected by a direct road, Dijkstra’s algorithm

can be used to find the shortest route between one city and all other cities.

Dijkstra’s Algorithm is the most common single-source shortest path algorithm. It require

three inputs (G, w, s) they are the graph G, the weights w, and the source vertex s. Graphs are often

used to model networks in which one travels from one point to another. A graph G (V, E) refers to a

collection of vertices V and a collection of edges E that connect pairs of vertices and assuming all

edge costs are non-negative, thus there are no negative cycles and shortest paths exists for all

vertices reachable from source vertex s. As a result, a basic algorithm problem is to determine the

shortest path between vertices in a graph.

Graphs G (V, E) are simple to define, the waypoints are the vertices V and there is edge E

between two waypoints, in the case of fuel optimal trajectory, the consumed fuel kdf from one

waypoint to another is the edge of these waypoints or vertices, and in the case of time optimal

trajectory, the travel time kd from one waypoint to another is the edge of these waypoints

(vertices). In both cases initial waypoint is the source vertex and the final waypoint is the destination

vertex.

https://en.wikipedia.org/wiki/Shortest-path_tree


14

2.1 Pseudo-code of Dijkstra’s Algorithm

To find the shortest path or to find the path with lowest cost between the source and destination

vertices, the Dijkstra’s algorithm must first initialize its three important arrays. First, the array S

contains the vertices that have already been examined or relaxed. It first starts as the empty set,

but as the algorithm progresses, it will fill with each vertex until all are examined. Then, the distance

array d[x] is defined to be an array of the shortest paths from source s to x. Finally, Q is simply the

data type used to form the list of all the vertices . In this case, it is a priority queue.

Now the algorithm moves into the shortest path calculation. The function will have to run as

long as it takes to relax each edge for each vertex. Next, the algorithm uses “ExtractMin” to extract

a vertex u from Q, this vertex u corresponding to the smallest shortest path estimate of any vertex

in Q, and then adds it to set S ( The first time though this loop, u = s). Then the algorithm compare

every edge that connects to this newly chosen vertex u. If the adjacent vertex v currently has a

distance to the source that is greater than the distance to u plus the distance between u and v, then

the algorithm update the distance to v. After completion of this step, we now have an array d[x] that

holds the value for the shortest distance from the source to each of the vertices in the graph. The

pseudo-code for Dijkstra’s algorithm is shown in (Figure 2-1).

Figure 2-1: Pseudo-code for Dijkstra’s algorithm


15

Figure 2-2: The execution of Dijkstra's algorithm

In (figure 2-2) a full example of the Dijkstra’s algorithm operation is shown. The source s is the

leftmost vertex. The shortest path estimates appears within the vertices, and shaded edges indicate

predecessor values. Black vertices are in the set S, and the white vertices are in the min-priority Q.

(a) The situation just before the first iteration of the while loop. The shaded vertex has the minimum

d value and is chosen as vertex u. (b) – (f) The situation after each successive iteration of the while

loop. The d value and predecessors shown in part (f) are the final values [14] [15] [16].

2.2 Representation of Graph

In order to input data into mathematics software in this case “Matlab”, it is important to have some

sort of method in which to describe the graph. One way to input the graph G into Matlab is in the

form of a square matrix. The matrix will always be of (n × n) dimension where n is equal to the

number of vertices in the graph. Each row will represent the vertex from which we are traveling.

Each column will represent the vertex to which we are traveling to.

The matrix G is a representation of the graph with three vertices. Vertex A corresponds to

row and column 1, B to row and column 2, etc. Matrix G shown below is the matrix representation of

the graph (figure 2-3).


16

0 1 2

G 1 0 3

2 3 0

It can now be seen that the cost from A to B will correspond to row 1, column 2 in the matrix.

Therefore, 𝐺 [1, 2] is equal to 1, 𝐺 [1, 3] is equal to 2, and 𝐺 [2, 3] is equal to 3. Also all of the

elements in the diagonal of the matrix are equal to 0. This is a result of describing the cost from one

vertex to itself, which is clearly zero. Also, the matrix should be symmetric across the diagonal. The

matrix being symmetric would imply 𝑤 (𝑥, 𝑦) = 𝑤 (𝑦, 𝑥) 𝑤ℎ𝑒𝑛 ∀𝑥, 𝑦 ∈𝐺.

2.3 Implementation of Dijkstra’s in Flight Trajectory Optimization

In addition to the basic formulation of the Dijkstra’s algorithm, the following aspects must be defined

specifically for the flight trajectory optimization problem. The number of vertices (waypoints) from

initial to final waypoints, the edge (consumed fuel or travel time ) between the vertices, defining

the source and destination vertices (waypoints). Once the above aspects have been accurately

defined, Dijkstra’s algorithm will determine the shortest path using the method described previously.

Figure 2-3: Representation of graph G into matrix

Chapter 3: Modeling of the 4D Waypoints Network

17

Chapter 3

3 Modeling of the 4D Waypoints Network

Waypoints are sets of coordinates that identify a point in physical space. It normally defined by

longitude , latitude , and altitude h . For 4D waypoint navigation it also defined by the arrival

time at that waypoint P .

Waypoints have only become widespread for navigational use by the layman since the

development of advanced navigational system, such as Global Positioning System (GPS) and certain

other type of radio navigation.

In the modern world, waypoints are increasingly abstract, often having no obvious relationship

to any distinctive features of the real world. These waypoints are used to help define invisible routing

paths for navigation. For example artificial airways created specifically for purpose of air navigation,

often have no clear connection to features of the real world, and consists only of a series of abstract

waypoint in the sky through which pilots navigates, these airways are designed to facilitate air traffic

control and routing of traffic between heavily traveled locations.

Suppose the waypoints network consists of N sets of waypoints, where 1P is the initial waypoint

and NP is the final waypoint. Each waypoint kP , (k =1,2,…., N) is defined by the geodetic coordinates

, ,k k kh , by considering the arrival time in each waypoint k , the waypoint kP can be described

as a four-dimensional state vector:

( , , , )T

k k k k kP h (3.1)

Where, k and k are the longitude and latitude of waypoint kP , kh is the altitude (with

respect to sea level). The following subsections represent the navigation model and constraints of 4D

waypoints network [17].


18

3.1 Navigation Model

The following differential equations model the dynamics of the navigation process:

cos sin

( )cose

V

R h

(3.2)

cos cos

( )e

V

R h

(3.3)

sinh V (3.4)

1V u (3.5)

2u (3.6)

3u (3.7)

Where, V = flight velocity, = flight path angle, =heading (with respect to the

geographical north), = longitude, = latitude, h = the altitude (with respect to sea level), eR is

the Earth radius. The variables 1u , 2u , and 3u are respectively the acceleration, the flight path angle

rate and the heading rate. The state vector x and control vector u of the above model are described

respectively as:

( , , , , , )Tx h V

1 2 3,( ), Tu u u u

3.2 Flight Constraints

The real world flight operate under several constraints, due to aircraft performance, aerodynamic

structural limits, safety reasons, the mission and other factor.

Velocity, V :

min max V V V (3.8)


19

Acceleration, a V :

min max a a a (3.9)

Flight path, :

min max (3.10)

Flight path angle rate, :

min max (3.11)

Heading rate, :

min max (3.12)

3.3 Arrival Time of Each Waypoint

The 4D navigation consists of traveling through a sequence of predefined point in a given time of

flight, which in turn define the trajectory of the flight. Assuming that the waypoint kP is already

defined by the geodetic coordinates longitude k , latitude k , and altitude kh , where the scheduled

time of arrival k at this waypoint is unknown. To compute this unknown arrival time of each

waypoint kP , the distance between this waypoint kP and its previous waypoint 1kP from where the

aircraft is arriving and the average velocity of the aircraft between this two waypoints are required.

The trajectory generation requires a geocentric coordinates system. To calculate the distance

between two waypoints, the 3D waypoint need to be transformed from usual geodetic coordinates

system to geocentric coordinates system. The 3D waypoint kP is defined by the following way:

( , , )T

k k k kP h (3.13)

Now to transform these geodetic coordinates to geocentric coordinates, the following

equation need to be applied [18].

( )cos cosk k k k kX N h (3.14)

( )cos sink k k k kY N h (3.15)


20

2[ (1 ) ]sink k k kZ N e h (3.16)

Being a is the Earth semi major axis and e its eccentricity, kN can be calculated as follows:

2 2

a

1 sink

k

Ne

(3.17)

After transforming the 3D waypoint from geodetic coordinates into geocentric coordinates

system, now it is possible to calculate kd (the distance between two waypoints

1kP to

kP ) by

using the following equation:

2 2 2

1 1 1(X ) (Y ) (Z )k k k k k k kd X Y Z (3.18)

To estimate the appropriate velocity of the aircraft kV at any waypoint kP , the following

equation can be used:

2k

k L

WV

C S (3.19)

Where, W is the aircraft nominal weight, LC is the lift coefficient, S is the wing area of

the aircraft and k is the air density (varies with altitude) at waypoint kP . It is possible to get the

appropriate velocity at any waypoint kP from Base of Aircraft Data (BADA), where true air speed,

TASV [kt] is specified for different aircraft for different flight level and phases of the flights [19].

By using the distance between two waypoints and the velocity of the aircraft in both

waypoints the arrival time of each waypoint can be computed as follow:

1

2

kk

k k

dd

V V

(3.20)

Where, kd is the time needed to go from waypoint 1kP to kP . So the arrival time of

waypoint kP from the initial waypoint can be described as follow:

1k k kd (3.21)

In practice, the aircraft may not pass through the waypoint kP exactly at this specified time

k due to disturbances. Therefore, an appropriate way is rather imposing a time tolerance in

(equation 3.21).


21

1k k kd (3.22)

Where, is the tolerance time interval for arrival at a determined waypoint [1.0 ≤ ≤ 1.4],

if the altitude of waypoints 1kP and kP are same, then the tolerance can be assumed 1.

3.4 Engine Thrust

To calculate the nominal fuel flow rate, it is necessary to calculate the engine thrust at different

phases of flight. The BADA model provides coefficients that allow the calculation of the following

thrust levels:

Maximum climb and take-off,

Maximum cruise,

Descent.

3.4.1 Maximum Climb and Take-off Thrust

The maximum climb thrust at standard atmosphere conditions, maxclimb( )ISAThr , is calculated as a

function of geo-potential pressure altitude, pH [ft]; true airspeed, TASV [kt]; and temperature

deviation from standard atmosphere, T [K]. Here only jet engine type was considered, The equation

for maximum climb and take-off thrust as follows:

,1 ,3

,2

2

maxclimb( )ISA= (1 )c c

c

p

T T p

T

HThr C C H

C (3.23)

Where, ,1cTC ,

,2cTC , and ,3cTC are climb thrust coefficient specified in the BADA tables.

The maximum climb thrust is corrected for temperature deviations from standard

atmosphere, T , in the following manner:

,5maxclimb maxclimb( )ISA (1 )

cT effThr Thr C T (3.24)


22

Where:

,4ceff TT T C (3.25)

With the limits:

,5

0.0 0.4cT effC T (3.26)

And:

,5

0.0cTC (3.27)

Where, ,4cTC , and

,5cTC are thrust temperature coefficient specified in the BADA tables. This

maximum climb thrust is used for both take-off and climb phases.

3.4.2 Maximum Cruise Thrust

The normal cruise thrust is by definition set equal to drag (Thr D ). However, the maximum

amount of thrust available in cruise situation is limited. The maximum cruise thrust is calculated as

a ratio of the maximum climb thrust as follows:

cruise maxclimb( )MAX

crTThr C Thr (3.28)

The Maximum cruise thrust coefficient crTC is currently uniformly set for all aircraft of

value 0.95.

3.4.3 Descent Thrust

Descent thrust is calculated as a ratio of the maximum climb thrust, with different correction factors

used for high and low altitudes, and approach and landing configurations, that is:

If ,p p desH H

des,highdes,high maxclimbTThr C Thr (3.29)

Where, ,p desH is the transition altitude,

des,highTC is high altitude descent thrust coefficient

specified in BADA tables.


23

If ,p p desH H

Cruise configuration:

des,lowdes,low maxclimbTThr C Thr (3.30)

Where, des,lowTC is the low altitude descent thrust coefficient specified in BADA tables.

Approach configuration:

des,appdes,app maxclimbTThr C Thr (3.31)

Where, des,appTC is the approach thrust coefficient specified in BADA tables.

Landing Configuration:

des,lddes,ld maxclimbTThr C Thr (3.32)

Where, des,ldTC is the landing thrust coefficient specified in BADA tables.

3.5 Fuel Consumption Model

This Subsection develops the fuel consumption model for commercial flights. In commercial flight

the rate of fuel burn depends on ambient temperature, aircraft speed, and aircraft altitude. It also

depend on the aircraft weight which changes as fuel burned. The Base of Aircraft Data (BADA) model

provides coefficients that allows to calculate the thrust specific fuel consumption and different

thrust level Thr , which can be used to calculate the nominal fuel flow rates nomf in different phases

of the flight.

3.5.1 Thrust Specific Fuel Consumption

For jet engine the thrust specific fuel consumption, [kg/ (min.KN)], is an engineering term that is

used to describe the fuel efficiency of an engine design with respect to thrust output, and is specified

as a function of the true airspeed, TASV [kt]:


24

1

2

(1 )TASf

f

VC

C (3.33)

Where, 1f

C , and 2f

C are the thrust specific fuel consumption coefficients specified in

BADA tables.

3.5.2 Nominal Fuel Flow Rate

The nominal fuel flow rate, nomf [kg/min], can then be calculated using the thrust:

nomf Thr (3.34)

The thrust varies with different flight phases, thus the thrust in this equation is depend on

the phase which the aircraft is flying, i.e. if the aircraft is flying in climb, cruise or descent phase

the thrust of this equation will be respectively climb, cruise or descent thrust. These expressions are

used in all flight phases except during idle descent and cruise, where, the following expressions are

to be used.

The minimum fuel flow rate, minf [kg/min], corresponding to idle thrust descent conditions,

is specified as a function of the geo-potential pressure altitude, pH [ft], that is:

3

4

min (1 )p

f

f

Hf C

C (3.35)

Where, 3f

C , and 4f

C are the descent fuel flow coefficients specified in BADA tables. The idle

thrust part of the descent stops when the aircraft switches to approach and landing configuration,

at which point thrust is generally increased. Hence, the calculation of the fuel flow during

approach and landing phases shall be based on the nominal fuel flow rate, and limited to the

minimum fuel flow:

/ minMAX( , )ap ld nomf f f (3.36)

The cruise fuel flow, crf [kg/min], is calculated using the thrust specific fuel consumption,

[kg/ (min.KN)], the thrust, Thr [N], and a cruise fuel flow correction coefficient, crfC :

crcr ff Thr C (3.37)


25

For the moment the cruise fuel flow correction factor has been established for a number of

aircraft types whenever the reference data for cruise fuel consumption is available. This factor has

been set to 1 (one) for all the other aircraft models

For now the nominal fuel flow rate nomf [kg/min], (equation 3.34) can be used for different

phase of the flight, as the cruise fuel flow correction factor crfC has been set to 1 for most of the

aircraft models, and if the thrust is not ideal during descent the fuel flow rate is based on the nominal

fuel flow rate [20].

3.6 Consumed Fuel Between Waypoints

BADA defines different flight phases for a departing trajectory, with specific performance values

to each phase. Thus the thrust, Thr [N] and the engine thrust specific fuel consumption, [kg/

(min.KN)], varies according to the condition verified at each flight moment. Which result different

nominal fuel flows rate, nomf [kg/min] for different phase of the flight. To generate a fuel optimal

trajectory from a set of waypoints in 4D waypoint network requires finding the associated fuel

consumed kdf by the aircraft to go from one waypoint to the other, defined as:

k nom kdf f d (3.38)

nomf [kg/min] is the nominal fuel flow rate, (equation 3.34). kdf [kg] is the amount of fuel

needed from waypoints 1kP to kP and, kd [min] is the amount of time needed to go from waypoints

1kP to kP . Which can be described in the following equations:

1k k kdf f f (3.39)

1k k kd (3.40)

Where, kf [kg] and k [min] are respectively the fuel and time required to get to waypoint

kP from initial waypoint , 1kf [kg] and 1k [min] are respectively the fuel and time required to get

to waypoint 1kP from initial waypoint.


26

Chapter 4: Optimal Trajectory Generation

27

Chapter 4

4 Optimal Trajectory Generation

To generate the fuel and time optimal trajectory, it is necessary to calculate the amount of consumed

fuel kdf and travel time kd needed between two waypoints, which was discussed In the previous

chapter. Then the Dijkstra’s algorithm was used to generate the fuel and time optimal trajectory by

inputting these data. To generate the fuel optimal trajectory the CI assumed to be zero and the time

cost was ignored and only amount of consumed fuel kdf to go from one waypoint to other was

considered. In the other case, to generate the time optimal trajectory, it was assumed that the CI

is maximum thus the fuel cost was ignored and only travel time kd between waypoints was

considered. The trajectories were generated in zero wind condition.

4.1 Selection of Flights for Analysis

For the analysis of this work short, medium and long haul 3 different flight lengths has been chosen.

They are:

Lisbon − Geneva

Lisbon − Stockholm

Lisbon – Montreal

Lisbon to Geneva is the short, Lisbon to Stockholm is the medium and Lisbon to Montreal is the

long haul flight. Three length of flights has been chosen to compare the result of fuel consumption.

The waypoints for each of the flight trajectories were chosen by the sky vector website [21]. The

initial and final waypoints for the each flight were chosen at altitude 3000 feet outside of the airport,

where the climb phase and landing approach begins. Firstly the waypoints were chosen by two

parameters (longitude and latitude ), then the altitude h of these waypoints were defined.

After having the tri dimensional waypoints the arrival time of each waypoint calculated as

described in the previous chapter. Different cruise altitude were chosen for different lengths of


28

flight, for short, medium and long haul flight the cruise altitudes are respectively 39000 feet, 41000

feet and 43000 feet.

In each flights the 4D waypoint network consists of two different trajectories, in each trajectory,

in short haul flight 10 waypoints, in medium haul flight 11 waypoints and in long haul flight 12

waypoints were chosen between the initial and final waypoints. Then the possible connection

between waypoints were established in these two trajectories for each flights.

4.2 Selection of Aircraft for Analysis

The selection of aircraft types is closely linked with the flight selection. The selection of cruise

altitude was also determined by the characteristics of the aircraft and the types of flight. In this

work we deal with two commercial aircraft A1 and A2. For the short haul flight Lisbon to Geneva,

the airplane A1 (which is a short to medium range twinjet narrow body airliner) was chosen, which

has 41,000 feet of maximum altitude and nominal weight of 60,000 kg. For the medium and long haul

flight Lisbon to Stockholm and Lisbon to Montreal, the airplane A2 (which is long range wide body

twinjet airliner), was chosen, which has 43,100 feet of maximum altitude and nominal weight of

211,000 kg. The characteristics of airplane A1 and airplane A2 are shown in table below.

Table 4-1: Characteristics of airplane A1 and airplane A2

Airplane A1 Airplane A2

Wing Area [m𝟐] 124.58 427.8

Maximum takeoff weight [kg] 70,800 287,000

Nominal weight [kg] 60,000 211,000

Cruise speed [Mach] 0.78 0.84

Maximum speed [Mach] 0.82 0.87

Maximum fuel capacity [L] 26,020 171,177

Maximum range [nm] 3,050 7,065

Engine x2 CFM 56-7 series GE 90-94B

Thrust x2 [KN] 121 417

Maximum altitude [ft] 41,000 43,100

For the estimation of fuel consumption in each flight the nominal weight of the aircraft was

considered.

https://en.wikipedia.org/wiki/Flight_length#Categories

https://en.wikipedia.org/wiki/Twinjet

https://en.wikipedia.org/wiki/Narrow-body_aircraft

https://en.wikipedia.org/wiki/Airliner


29

4.3 Fuel Optimal Trajectory Generation

The CI (cost index) is zero in maximum range airspeed and minimum trip fuel. This speed schedule

ignores the cost of time. To generate the fuel optimal trajectory, the consumed fuel kdf between

waypoints were used in Dijkstra’s algorithm. When trying to find a shortest (lowest cost) path for an

aircraft by using Dijkstra’s algorithm between two given points in space, the first step is to build a

graph.

Graphs are often used to model network in which one travel from one point to another. A

graph G(V, E) contains all vertices V and edges E that connect pairs of vertices. In this case of fuel

optimal trajectory the waypoints are the vertices V and the consumed fuel kdf between the

waypoints are the edges E.

As two different trajectories were chosen in each flight it is possible to establish connection

between waypoints in these two trajectories. Once the connection has been established it is possible

to calculate the consumed fuel kdf between the vertices (waypoints), which is edges between them.

Then the edge ( kdf ) between all the vertices (waypoints) are used to build up the graph, by using

this graph to Dijkstra’s shortest path algorithm, the fuel optimal trajectory was generated.

4.4 Time Optimal Trajectory Generation

The maximum value for CI (cost index) uses a minimum time speed schedule. This speed schedule

calls for maximum flight envelope speeds, and ignore the cost of fuel. To generate the time optimal

trajectory it is needed to build a graph same as to generate the fuel optimal trajectory, but in the

case of time optimal trajectory the edge of the graphs are the associated travel time kd between

the waypoints instead of consumed fuel kdf .

First the associated travel time kd between the waypoints were calculated in each of these

two trajectories, then the possible connection of waypoints between these two trajectories were

established and associated travel time kd between them was calculated. After that the graph was

built by these vertices (waypoints) and edges ( kd ). The Dijkstra’s algorithm use this graph to find

the time optimal trajectory from initial waypoint to its final waypoint.


30

4.5 Minimization of Delay of each Waypoint in Fuel and Time Optimal

Trajectories.

Once the fuel or time optimal trajectory is generated, it is possible to minimize the delay of each

waypoint in the whole trajectory. Considering the trajectory have n sequence of waypoints in 4D,

1 1 1 1 1(X ,Y ,Z , )TP , 2 2 2 2 2(X ,Y ,Z , )TP ,………….. , (X ,Y ,Z , )T

n n n n nP where, X , Y , Z

are the geocentric coordinates of the waypoints, and is the arrival time of that waypoints.

To minimize the delay at each waypoints, the trajectory is divided into segments, each

segments consist of 3 waypoints, assuming the number of segment is k [k = 1,------- , (n − 2)], the

segment k consist of these waypoints, kP , 1kP and 2kP for time ∈ [ k , 1k and 2k ]. The

geocentric coordinates X , Y , Z for different waypoints can be expressed by the following way for

k segment:

2 3

0 1 2 3( ) ( ) ( ) ( )k k k k k

k k kX a a a a (4.1)

2 3

0 1 2 3( ) ( ) ( ) ( )k k k k k

k k kY b b b b (4.2)

2 3

0 1 2 3( ) ( ) ( ) ( )k k k k k

k k kZ c c c c (4.3)

Where, ∈ [ k , 1k and 2k ].

Using the above equations for (n − 2) segments, it is possible to minimize the delay of each

waypoints for the whole trajectory. In equations (4.1), (4.2) and (4.3) the unknowns are respectively

0a , 1a , 2a , 3a for X coordinate, 0b , 1b , 2b , 3b for Y coordinate and 0c , 1c , 2c , 3c for Z coordinate.

Once all the unknowns are calculated, by using these unknown coefficient it is possible to simulate

the trajectory by minimizing delay in each waypoints.

4.5.1 Determination of coefficients

The unknown coefficients can be determined by the following way: For segment k=1, and = 1 ,

from equation (4.1):

1 1 1 1 2 1 3

1 0 1 1 1 2 1 1 3 1 1( ) ( ) ( ) ( )X a a a a (4.4)

So,


31

1 0X a (4.5)

Now, the first derivatives of equation (4.4) is

1 1 1 1 2

1 1 2 1 1 3 1 1( ) 2 ( ) 3 ( )X a a a (4.6)

1 1dX a (4.7)

Where, the initial slope of 1X to 2X is the following equation:

2 11

2 1

X XdX

(4.8)

The value of coefficients 0a and 1a is found From equation (4.5) and (4.7). Now for the

segment k=1 of X coordinates the unknowns are 2a and 3a , for segment k=1 and for = [ 2 , 3 ],

equation (4.1) can be written as:

1 1 1 1 2 1 3

2 0 1 2 1 2 2 1 3 2 1 2( ) ( ) ( ) ( )X a a a a X (4.9)

and,

1 1 1 1 2 1 3

3 0 1 3 1 2 3 1 3 3 1 3( ) ( ) ( ) ( )X a a a a X (4.10)

Now by solving the equation (4.9) and (4.10), it is possible get the value of coefficient 2a

and 3a . After having all the coefficient for X coordinate for segment k=1 it is possible to simulate

the trajectory of X coordinate of this segment.

To get the unknown coefficients of Y and Z coordinates in segment k=1, the same approach

can be used like the X coordinates. Now calculating the unknown coefficients of X , Y , Z for all

the segments k = [1,--------, (n − 2)], it is possible to simulate the whole trajectory by minimizing the

delays in each waypoint.


32

Chapter 5: Simulation and Result

33

Chapter 5

5 Simulation and Result

In this chapter the simulation and result of the fuel and time optimal trajectories are shown for three

different length (short, medium and long haul) trajectories considering no wind. All the analysis of

the simulation has been done using Matlab 2013a. The coordinates of the trajectory were chosen using

skyvector website [21].

5.1 Short Haul Flight

To analyze the short haul flight, the flight from Lisbon to Geneva was considered. The initial waypoint

is in 3000 feet and 14 nm away from Lisbon airport, where climb phase begins. The final waypoint

also is in 3000 feet and 26 nm away from Geneva airport, where the landing approach begins.

The 4D waypoints network of this short haul flight consists of two trajectories, which has

total 22 waypoints including the initial and final waypoints, and each trajectory has 12 waypoints

including the initial and final waypoints. Airplane A1 was used to analyze the flight trajectories. The

cruise altitude of the both trajectories is at 39000 feet. (table 5-1 and 5-2) Show the waypoints lists

for both of the trajectories. Each waypoint is defined in geodetic coordinates ( , , h ) and their

associated distance kd , travel time kd and consumed fuel kdf between the waypoints are also

shown.

The trajectories were chosen such a way that the climb and descent phases of the first

trajectory are smaller than the second trajectory but the cruise phase of the first trajectory is bigger

than the second trajectory, therefore the total distance from the initial waypoint to final waypoint

for both of the trajectories are more or less same.

In order to find the fuel and time optimal trajectory from the 4D waypoint network possible

connection between waypoints in both trajectories were established, and their associated distance

kd , travel time kd and consumed fuel kdf between these possible waypoints connections were

calculated.


34

Table 5-1: List of waypoints in 1st trajectory of short haul flight

waypoint [deg]

[deg] h [feet]

kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -8.9083 39.1087 10000 9.249827 2.371751 291.0612

P3 -8.624 39.33417 20000 19.01912 3.437191 348.0156

P4 -7.7987 39.8783 33000 50.39803 7.295251 535.3803

P5 -6.9993 40.2513 39000 43.16538 5.748997 294.6361

P6 -3.3707 43.227 39000 242.0397 32.48854 1176.085

P7 0.1303 44.729 39000 176.7117 23.71969 858.6528

P8 3.5963 44.9543 39000 148.8367 19.97808 723.2065

P9 3.87217 45.1205 33000 15.45767 2.058735 7.823191

P10 4.6985 45.41983 20000 39.46054 5.71202 32.05871

P11 5.336 45.606 10000 29.19265 5.275779 42.20624

Final (P22) 5.7553 45.884 3000 24.30017 5.618537 54.83692

Total 797.8314 113.7046 4363.963

Table 5-2: List of waypoints in 2nd trajectory of short haul flight

waypoint [deg]

[deg] h [feet]

kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P12 -8.835 39.0883 10000 11.16491 2.862798 351.3225

P13 -8.49983 39.30183 20000 20.28586 3.66612 371.1947

P14 -7.59217 39.7763 33000 50.94383 7.374258 541.1783

P15 -6.715 40.11317 39000 45.32355 6.036433 309.3672

P16 -1.765 41.631 39000 243.352 32.6647 1182.462

P17 0.5565 43.9993 39000 175.4797 23.55432 852.6664

P18 3.6277 44.85317 39000 141.8812 19.04446 689.4095

P19 3.9993 44.9983 33000 18.13834 2.415762 9.179894

P20 4.8617 45.261 20000 39.98448 5.787862 32.48437

P21 5.4975 45.5203 10000 31.13026 5.625951 45.00761

Final (P22) 5.7553 45.884 3000 24.40439 5.642634 55.0721

Total 802.0885 114.6753 4439.345


35

5.1.1 Fuel Optimal Trajectory

The fuel optimal trajectory was generated from the 4D waypoints network using the Dijkstra’s

shortest path algorithm, the associated consumed fuel kdf between waypoints were used as edges

to find the fuel optimal trajectory.

The fuel optimal trajectory contains 9 waypoints [initial waypoint (P1)→ P2→ P3→ P4→ P5→

P18→ P19→ P11→ final waypoint (P22)], the distance between the initial and final waypoints in fuel

optimal trajectory is 777.7 nm. The comparison of consumed fuel in different phases of flight for

these two trajectories and fuel optimal trajectory are shown in (table 5-3).

Table 5-3: Total fuel consumed in different trajectories for short haul flight.

Trajectory Fuel consumed [kg] Total [kg]

Climb Cruise Descent

1 1469.1 2757.9 136.9 4363.9

2 1573.1 2724.5 141.7 4439.3

Fuel optimal 1469.1 2652.8 136.1 4258

As seen in the (table 5-3), by using the fuel optimal trajectory for the short haul flight (Lisbon

– Geneva) the aircraft consumes 105.9 kg of less fuel than the first trajectory, which is equivalent to

2.4% less fuel than the first trajectory and consumes 181.3 kg of less fuel than the second trajectory,

which is equivalent to 4.1% less fuel than the second trajectory. The fuel optimal trajectory in 3D is

shown in (figure 5-2).

Figure 5-1: 3D fuel optimal trajectory in geocentric coordinates for short haul flight.


36

In figure (5-1) the curve blue line represents the real trajectory path through the different

waypoints, starting on the right. The red circle around the blue line denote the position of the original

waypoints.

5.1.2 Time Optimal Trajectory

The time optimal trajectory was also generated from the 4D waypoints network by using the Dijkstra’s

shortest path algorithm. Possible connections of waypoints between these trajectories in the

waypoint network were established, then the associated time between these waypoints were used

to find the time optimal trajectory.

The time optimal trajectory contains 10 waypoints [initial waypoint (P1)→ P2→ P3→ P4→ P5→

P8→ P9→ P10→ P11→ final waypoint (P22)], the distance between the initial and final waypoints in the

time optimal trajectory is 777.8 nm. The comparison of travel time in different phases of flight for

those two trajectories and time optimal trajectory are shown in (table 5-4).

Table 5-4: Total time needed in different trajectories for short haul flight.

Trajectory Time [min] Total [min]


1 18.9 76.2 18.7 113.7

2 19.9 75.3 19.5 114.7

Time optimal 18.9 73.49 18.7 111.01

Figure 5-2: 3D time optimal trajectory in geocentric coordinates for short haul flight.


37

As it is seen from (table 5-4) by using the time optimal trajectory in short haul flight (Lisbon

– Geneva), the aircraft reaches the final waypoint 2.7 minutes faster than the first trajectory, which

is equivalent to 2.4% of total travel time of the first trajectory and 3.7 minutes faster than the second

trajectory equivalent to 3.2% of total travel time of the second trajectory. The time optimal

trajectory is shown in (figure 5-2), the time optimal trajectory is represented by the blue line and

the red circles around the trajectories are the waypoints.

5.2 Medium Haul Flight

The flight from Lisbon to Stockholm was considered to analyze the medium haul flight. The initial

waypoint is in 3000 feet and 14 nm away from Lisbon airport, where climb phase begins. The final

waypoint is also in 3000 feet and 19 nm away from Stockholm Arlanda airport, where the landing

approach begins.

There are two trajectories between the initial and final waypoints in the 4D waypoints

network, each trajectories has 13 waypoints including the initial and final waypoints, and total 24

waypoints are there in the waypoint network including the initial and final waypoint. Airplane A2 was

used to analyze the flight trajectories. The cruise altitude of the both trajectories is at 41000 feet.

(table 5-5 and 5-6) Show the waypoints lists for both of the trajectories.

Table 5-5: List of waypoints in 1st trajectory of medium haul flight

waypoint [deg]

[deg] h [feet]

kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -8.9373 39.1525 10000 10.63985 2.665515 1070.631

P3 -8.643 39.6385 24000 32.29915 5.529098 1724.94

P4 -8.0007 40.5407 37000 61.77433 8.291856 1859.863

P5 -7.7847 40.942 41000 26.05963 3.243938 589.0991

P6 -4.752 46.1707 41000 340.8976 42.43539 5385.051

P7 1.07817 49.6497 41000 314.9914 39.21055 4975.819

P8 9.07617 53.515 41000 379.0242 47.18143 5987.324

P9 14.4575 57.839 41000 318.0823 39.59531 5024.645

P10 14.75417 58.0005 37000 13.62262 1.695762 31.54117

P11 15.78983 58.624 24000 49.92285 6.807662 161.8522

P12 17.0113 59.2063 10000 51.77227 9.043191 287.5735

Final (P24) 17.73983 59.3445 3000 23.95284 5.538229 212.0034

Total 1623.039 211.2379 27310.34


38

Table 5-6:List of waypoints in 2nd trajectory of medium haul flight

waypoint [deg]

[deg] h [feet]

kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P13 -8.8405 39.1047 10000 11.47263 2.874146 1154.43

P14 -8.455 39.576 24000 33.57806 5.748028 1793.241

P15 -7.808 40.4545 37000 60.65319 8.141368 1826.109

P16 -7.413 40.8377 41000 29.26358 3.642769 661.5269

P17 -0.2773 44.76 41000 393.6387 49.00066 6218.184

P18 4.6187 50.023 41000 374.0856 46.56667 5909.311

P19 10.92483 54.85883 41000 371.7582 46.27695 5872.545

P20 15.0405 57.2845 41000 201.3458 25.06379 3180.595

P21 15.39 57.4757 37000 16.19891 2.016462 37.50619

P22 16.496 58.17 24000 54.91818 7.488843 178.0473

P23 17.504 58.94117 10000 56.26139 9.827318 312.5087

Final (P24) 17.73983 59.3445 3000 25.36559 5.864875 224.5074

Total 1628.54 212.5119 27368.51


The fuel optimal trajectory contains 9 waypoints [initial waypoint (P1)→ P2→ P3→ P4→ P5→

P20→ P21→ P23→ final waypoint (P24)], the distance between the initial and final waypoints in fuel

optimal trajectory for this flight is 1596.5 nm. The comparison of consumed fuel in different phases

of flight for different trajectories including the fuel optimal trajectory are shown in (table 5-7).

Table 5-7: Total Fuel consumed in different trajectories for medium haul flight.



1 5244.5 21372.8 692.97 27310.3

2 5435.3 21180.6 752.6 27368.5

Fuel optimal 5244.5 20744.4 742.2 26731.1

From the initial waypoint to reach the final waypoint using the fuel optimal trajectory the

aircraft consumes 579.2 kg of less fuel than the first trajectory and consumes 637.4 kg of less fuel

than the second trajectory. In other word by using the fuel optimal trajectory for the medium haul


39

flight (Lisbon – Stockholm) the aircraft consumes 2.1% less fuel than the first trajectory, and 2.3%

less fuel than the second trajectory. The fuel optimal trajectory is shown in (figure 5-4).

The blue curve in (figure 5-3) corresponds to the fuel optimal trajectory for the medium haul

flight (Lisbon – Stockholm) and the red circles around the curve are the waypoints of the fuel optimal

trajectory.


The time optimal trajectory for the flight between Lisbon and Stockholm contains 9 waypoints

[initial waypoint (P1)→ P2→ P3→ P4→ P5→ P9→ P10→ P12→ final waypoint (P24)], the distance between

the initial and final waypoints in time optimal trajectory is 1589.6 nm. The comparison of travel time

in different phases of the flight for those two trajectories and the time optimal trajectory are shown

in (table 5-8).

Table 5-8: Total time needed in different trajectories for medium haul flight.



1 19.7 168.4 23.1 211.2

2 20.4 166.9 25.2 212.5

Time optimal 19.7 164.3 23.04 207.04

Figure 5-3: 3D fuel optimal trajectory in geocentric coordinates for medium haul flight.


40

By flying the time optimal trajectory in medium haul flight (Lisbon – Stockholm) 4.2 minutes

or 1.9% of total travel time can be saved than the first trajectory and 5.5 minutes or 2.6% of total

travel time can be saved than the second trajectory. The time optimal trajectory in 3D is shown in

(figure 5-4).

The blue curved line is the time optimal trajectory path and the red circles around it are the

associated waypoints of the time optimal trajectory.

5.3 Long Haul Flight

The flight from Lisbon to Montreal was considered to analyze the long haul flight. The initial waypoint

is in 3000 feet and 14 nm away from Lisbon airport, where climb phase begins. The final waypoint is

also in 3000 feet and 18 nm away from Montreal airport, where the landing approach begins.

The 4D waypoints network of the long haul flight consists of two trajectories between the

initial and final waypoint, each trajectories has 14 waypoints including the initial and final waypoints,

total 26 waypoints in the whole waypoint network including initial and final waypoints. Airplane A2

was used to analyze the flight trajectories. The cruise altitude of the both trajectories is at 43000

feet. The waypoints lists of both trajectories are shown in (table 5-9 and 5-10).

Figure 5-4: 3D time optimal trajectory in geocentric coordinates for medium haul flight.


41

Table 5-9: List of waypoints in 1st trajectory of long haul flight

waypoint [deg]

[deg] h [feet]

kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -9.22083 39.1 10000 10.5683 2.64759 1063.431

P3 -9.8425 39.3857 24000 33.76592 5.780186 1803.274

P4 -10.9352 39.9543 37000 61.15747 8.209056 1841.291

P5 -11.74 40.32817 43000 43.3812 5.40015 959.6066

P6 -22.5022 43.5023 43000 518.4016 64.53132 8227.743

P7 -35.132 44.73617 43000 550.8432 68.56969 8742.636

P8 -44.9847 47.4295 43000 442.3112 55.05948 7020.084

P9 -58.667 48.5647 43000 555.3268 69.12781 8813.796

P10 -70.3565 46.4947 43000 491.3069 61.15854 7797.714

P11 -70.809 46.4135 37000 19.45513 2.421801 43.59241

P12 -71.9328 46.21 24000 48.42787 6.6038 157.0053

P13 -73.0908 45.9335 10000 51.23244 8.948898 284.575

Final (P26) -73.6412 45.7947 3000 24.56775 5.680405 217.4459

Total 2850.746 364.1387 46972.19

Table 5-10: List of waypoints in 2nd trajectory of long haul flight

waypoint [deg]

[deg]

h [feet]

kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P14 -9.2387 39.05683 10000 10.03791 2.514717 1010.061

P15 -9.94583 39.2603 24000 35.28932 6.040968 1884.631

P16 -11.2588 39.4515 37000 62.29715 8.362034 1875.604

P17 -11.9673 39.9493 43000 44.44929 5.533107 983.2331

P18 -24.8818 42.095 43000 600.2876 74.7246 9527.386

P19 -34.374 43.996 43000 433.2285 53.92886 6875.93

P20 -44.3497 46.44083 43000 448.0273 55.77103 7110.806

P21 -61.77 47.44 43000 717.2308 89.28184 11383.43

P22 -70.163 46.2543 43000 353.4368 43.99628 5609.525

P23 -70.557 46.102 37000 18.61899 2.317716 41.71889

P24 -71.739 45.9043 24000 51.21564 6.98395 166.0434

P25 -72.9952 45.81 10000 53.07175 9.270174 294.7915

Final (P26) -73.6412 45.7947 3000 27.16449 6.280807 240.4293

Total 2854.355 365.0061 47003.6


42


The fuel optimal trajectory of the flight between Lisbon and Montreal contains 9 waypoints [initial

waypoint (P1)→ P14→ P3→ P4→ P5→ P22→ P11→ P13→ final waypoint (P26)], the distance between the

initial and final waypoints in fuel optimal trajectory is 2777 nm. The comparison of consumed fuel in

different phases of flight for different trajectories and the fuel optimal trajectory are shown in (table

5-11).

Table 5-11: Total Fuel consumed in different trajectories for long haul flight.



1 5667.6 40601.9 702.6 46972.2

2 5753.5 40507.1 742.98 47003.6

Fuel optimal 5652.1 39284.9 712.5 45649.5

Using the fuel optimal trajectory in long haul flight (Lisbon – Montreal) the aircraft consumed

1322.7 kg of less fuel than the first trajectory and consumed 1354.1 kg less fuel than the second

trajectory. So in another word by flying the fuel optimal trajectory for the long haul flight the aircraft

consumes 2.8% less fuel than the first trajectory, and 2.9% less fuel than the second trajectory. The

fuel optimal trajectory in 3D is shown in (figure 5-5).

The blue curve in the (figure 5-5) is the fuel optimal trajectory and the red circles around it

are the waypoints of that trajectory. In this long haul flight the cruise phase is really large compare

Figure 5-5: 3D fuel optimal trajectory in geocentric coordinates for long haul flight.


43

to its climb and descent phases, thus in the figure the waypoints in the climb and descent phases are

seems too close to each other.


The time optimal trajectory of flight between Lisbon to Montreal contains 9 waypoints [initial

waypoint (P1)→ P14→ P3→ P4→ P5→ P10→ P11→ P13→ final waypoint (P26)], the distance between the

initial and final waypoints in time optimal trajectory of this flight is 2772 nm. The comparison of

travel time in different phases of flight for different trajectories and time optimal trajectory are

shown in (table 5-12).

Table 5-12: Total time needed in different trajectories for long haul flight.



1 22.04 318.5 23.7 364.2

2 22.5 317.7 24.9 365.1

Time optimal 22.03 308.6 23.6 354.3

By using the time optimal trajectory in long haul flight (Lisbon – Montreal) the aircraft reaches

to the final waypoint from the initial waypoint 9.9 minutes faster than the first trajectory which

Figure 5-6: 3D fuel optimal trajectory in geocentric coordinates for long haul flight.


44

saves 2.7% of the total travel time and 10.8 minutes faster than the second trajectory which save

2.9% of total travel time. The time optimal trajectory in 3D is shown in (figure 5-6).

In the (figure 5-6) the blue curve is the time optimal trajectory and the red circles around it

are the waypoints of the trajectory. In the time optimal trajectory the cruise phase of the flight is

really large compare to its climb and descent phases, thus in the figure the waypoints in the climb

and descent phases are seems too close to each other.

Chapter 6: Conclusion

45

Chapter 6

6 Conclusion and Discussions

This study is based on finding the fuel and time optimal trajectories of climb, cruise and descent

phases of the flight, without the takeoff and landing phases of the flight. In this work several steps

were made in order to achieve a complete trajectory from 4D waypoint networks that minimizes the

fuel consumption and travel time. This study uses Dijkstra’s shortest path algorithm that finds the

shortest path in a graph, which is refers to a collection of vertices (waypoints) and a collection of

edges (associated consumed fuel or travel time) that connect the pairs of vertices. This technique

was used to compare different length (short, medium and long haul) flights.

The analysis results show promising potential for reduction of consumed fuel and travel time

in different flights via using the Dijkstra’s shortest path algorithm, across a range of common aircraft

and routes. The results suggest that by flying fuel and time optimal trajectory for short haul flight,

it is possible to save 2.4−4.1% on fuel burn, which is equivalent to 105.9 – 181.3 kilograms of fuel

and 2.7−3.7 minutes or 2.4 - 3.2 % of total travel time. In medium haul flight by flying the fuel

optimal trajectory can potentially save 2.1−2.3% fuel, reducing fuel burn by 579.2 – 637.4 kilograms

and by flying the time optimal trajectory the travel time was reduced by 4.2−5.5 minutes or 1.9 –

2.6% of total travel time. For long haul flight it is possible to save 2.8−2.9% on fuel burn, which is

equivalent to 1322.7 – 1354.1 kilogram of fuel by flying the fuel optimal trajectory, and 9.9−10.8

minutes or 2.7 -2.9% of total travel time was saved by flying the time optimal trajectory. In general

the savings of the fuel and time are proportional to the trip lengths, and depends on the aircraft

types.

The results show that during the cruise phase, the aircraft consumes the majority of fuel but

fuel consumption per unit time is higher during the climb phase and in descent phase the fuel

consumption per unit time is smaller than the other two phases. So for the fuel optimal trajectory

the algorithm tries to obtain the trajectory by choosing short climb and cruise phases and long

descent phase.

Chapter 6: Conclusion

46

6.1 Future work

Although the algorithm has proven reliable to find the fuel and time optimal trajectory from 4D

waypoint networks, there is still room for improvement. By using more trajectories and networks

with different cruise altitude more fuel and time can be saved, as there will be more waypoints to

choose from.

In addition, a realistic wind model and Air traffic control (ATC) restrictions can be imposed

on the 4D waypoints network to find the fuel and time optimal trajectory in presence of wind and

other source of disturbances to model more realistic flight.

47

7 References

[1] ICAO, "ICAO Environmental Report: Aviation and Climate change," ICAO, 2013.

[2] Penner J. E, Lister D. H, Griggs D. J, Dokken D. J and McFarland M, "Aviation and the Global

Atmosphere," Intergovernmental Panel on Climate Change, 1999.

[3] S. W. airlines, "Fuel Savings: Reducing Our Environmental Impact," [Online]. Available:

https://www.southwest.com/html/southwest-difference/southwest-

citizenship/environmental-initiatives/. [Accessed June 2015].

[4] C. P. airlines, "Cathay Pacific “silver bullet” freighter takes to the skies," [Online]. Available:

http://centreforaviation.com/analysis/cathay-pacific-silver-bullet-freighter-takes-to-the-

skies-4553. [Accessed June 2015].

[5] Roberson B, "Fuel Conservation Strategies: Cost Index Explained," Boeing, 2007, pp 26-28.

[6] Airbus, "Getting to grips with fuel economy," Airbus, October 2004.

[7] Roberson W, Root R and Adams D, "Fuel Conservation strategies : Cruise Flight," Boeing, 2007,

pp 23-27.

[8] Roberson W and Johns J. A, "Fuel Conservation Strategies: Takeoff and Climb," Boeing, 2008,

pp 25-28.

[9] Allison R. L, "Application of NASA-ARC delayed flap approach procedure to Boeing 727 Airplane,"

NASA, February 1977.

[10] Roberson W and Johns j. A, "Fuel Conservation Stategies: Descent and Approach," Boeing, 2008,

pp 25-28.

[11] Bryson A. E and Ho Y. C, "Applied Optimal Control: Optimization, Estimation and, Control",

Taylor & Francis, New York, 1975, pp. 42-89.

[12] Calise A," Extended Energy Management for Flight Performance Optimization", AIAA Journal,

VOl. 15, No. 3, March 1977, pp. 314-321.

[13] Calise A and Moerder D, "Singular Perturbation Techniques for Real Time Aircraft Trajectory

Optimization and Control", NASA CR-3597, 1982.

[14] Cormen T. H, Leiserson C. E, Rivest R. L and Stein C, "Introduction to algorithms", London,

England: The MIT press, 2009, pp. 658-659.

[15] Hart C, "Graph Theory Topics in Computer Networking," 2013, pp 13-20.

[16] Dasgupta S, Papadimitriou C. H and Vazirani U. V, "Algorithms", McGraw-Hill, New York, July

18, 2006, pp. 112-118.

48

[17] Machado P and Bousson K, "4D Flight Trajectory Optimization Based on Pseudospectral

Methods," WSEAS transactions on system and control, Vol 8, No 3, july 2013, pp 105-119.

[18] Seemkooei A. A, "Comparison of different algorithm to transform geocentric to geodetic

coordinates," Survey Review 36, October 2002, pp 627-632.

[19] EUROCONTROL "Aircraft Performance Summary Tables for the Base of Aircraft Data (BADA),"

Eurocontrol Experimental Centre, Revision 3.4, June 2002,.

[20] EUROCONTROL, "User manual for the base of aircraft data (BADA) revision 3.9," Eurocontrol

Experimental Centre, Revision 3.4, April 2011.

[21] "SkyVector," [Online]. Available: https://skyvector.com/. [Accessed October 2015].

Annex A – Results

49

Annex A – Results

Short Haul Flight

Fuel Optimal Trajectory

waypoint [deg]

[deg] h

[feet] kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -8.9083 39.1087 10000 9.24982731 2.371750592 291.0612327

P3 -8.624 39.33417 20000 19.01912267 3.437190843 348.0155729

P4 -7.7987 39.8783 33000 50.39802743 7.295251257 535.3802516

P5 -6.9993 40.2513 39000 43.16538208 5.748996503 294.6360708

P18 3.6277 44.85317 39000 545.9488459 73.28172428 2652.798419

P19 3.9993 44.9983 33000 18.13834289 2.41576154 9.179893851

P11 5.336 45.606 10000 67.50872653 10.90315906 72.04807509

Final (P22) 5.7553 45.884 3000 24.3001715 5.618536764 54.83691882

Total 777.7284463 111.0723708 4257.956435

Time Optimal Trajectory

waypoint [deg]

[deg] h

[feet] kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -8.9083 39.1087 10000 9.24982731 2.371750592 291.0612327

P3 -8.624 39.33417 20000 19.01912267 3.437190843 348.0155729

P4 -7.7987 39.8783 33000 50.39802743 7.295251257 535.3802516

P5 -6.9993 40.2513 39000 43.16538208 5.748996503 294.6360708

P8 3.5963 44.9543 39000 547.5123329 73.49158831 2660.395497

P9 3.87217 45.1205 33000 15.45766508 2.058734528 7.823191205

P10 4.6985 45.419833 20000 39.46053795 5.712019968 32.05871207

P11 5.336 45.606 10000 29.19264614 5.275779423 42.20623539

Final (P22) 5.7553 45.884 3000 24.3001715 5.618536764 54.83691882

Total 777.7557131 111.0098482 4266.413682

Annex A – Results

50

Medium Haul Flight


waypoint [deg]

[deg] h

[feet] kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -8.9373 39.1525 10000 10.63984735 2.665514994 1070.630752

P3 -8.643 39.6385 24000 32.2991454 5.529097643 1724.940237

P4 -8.0007 40.5407 37000 61.77432604 8.291855844 1859.863266

P5 -7.7847 40.942 41000 26.05963408 3.243937852 589.099114

P20 15.0405 57.2845 41000 1313.210885 163.4702346 20744.37277

P21 15.39 57.4757 37000 16.19890882 2.016461679 37.50618723

P23 17.504 58.94117 10000 110.97267 17.27201089 480.1619027

Final (P24) 17.73983 59.3445 3000 25.36558534 5.864875223 224.5074236

Total 1596.521002 208.3539887 26731.08165


waypoint [deg]

[deg] h

[feet] kd

[nm] kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -8.9373 39.1525 10000 10.63984735 2.665514994 1070.630752

P3 -8.643 39.6385 24000 32.2991454 5.529097643 1724.940237

P4 -8.0007 40.5407 37000 61.77432604 8.291855844 1859.863266

P5 -7.7847 40.942 41000 26.05963408 3.243937852 589.099114

P9 14.4575 57.839 41000 1319.621296 164.2682112 20845.636

P10 14.75417 58.0005 37000 13.62262133 1.695761991 31.54117304

P12 17.0113 59.2063 10000 101.5868205 15.8111783 439.5507566

Final (P24) 17.73983 59.3445 3000 23.95284067 5.538229057 212.0034083

Total 1589.556532 207.0437869 26773.26471

Annex A – Results

51

Long Haul Flight


waypoint [deg]

[deg] h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P14 -9.2387 39.05683 10000 10.0379116 2.514716893 1010.061187

P3 -9.8425 39.3857 24000 34.47508776 5.90158421 1841.146734

P4 -10.93517 39.9543 37000 61.15746523 8.209055735 1841.291201

P5 -11.74 40.32817 43000 43.38120141 5.400149554 959.6065757

P22 -70.163 46.2543 43000 2475.208848 308.117284 39284.95371

P11 -70.809 46.4135 37000 28.56382277 3.555662585 64.00192654

P13 -73.09083 45.9335 10000 99.61194468 15.50380462 431.0057684

Final (P26) -73.64117 45.7947 3000 24.56775255 5.680405215 217.4459116

Total 2777.004034 354.8826628 45649.51302


waypoint [deg]

[deg] h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P14 -9.2387 39.05683 10000 10.0379116 2.514716893 1010.061187

P3 -9.8425 39.3857 24000 34.47508776 5.90158421 1841.146734

P4 -10.93517 39.9543 37000 61.15746523 8.209055735 1841.291201

P5 -11.74 40.32817 43000 43.38120141 5.400149554 959.6065757

P10 -70.3565 46.4947 43000 2479.302175 308.6268268 39349.92041

P11 -70.809 46.4135 37000 19.45513208 2.421800673 43.59241212

P13 -73.09083 45.9335 10000 99.61194468 15.50380462 431.0057684

Final (P26) -73.64117 45.7947 3000 24.56775255 5.680405215 217.4459116

Total 2771.98867 354.2583437 45694.0702

Annex A – Results

52

Annex B – Article

53

Annex B – Article “Optimal Fuel Conservation in

4D Waypoint Networks”

(Submitted to a refereed journal listed in the ISI Journal Citation Reports)

Annex B – Article

54

Annex B - Article

55

Optimal Fuel Conservation in 4D Waypoint Networks

Kawser Ahmed1

Department of Aerospace Science, University of Beira Interior, Covilhã, Portugal

Abstract

The purpose of this work is to develop a trajectory optimization algorithm that finds a fuel optimal

trajectory from a 4D waypoint networks, where the arrival time is specified for each waypoint in the

networks. A single source shortest path algorithm is presented to generate the optimal aircraft trajectory

that minimizes fuel burn, generating such trajectory helps the aviation industry cope with increasing

fuel costs and reduce aviation induced climate change, as CO2 is directly related to the amount of fuel

burned, reduction in fuel burn implies a reduction in carbon emissions as well. Results of this work

shows that by flying a fuel optimal trajectory, which was found by implying a single source shortest

path algorithm (Dijkstra’s algorithm) can lead to reduction of average fuel burn of international flights

by 2.8% of the total trip fuel.

Key words: Fuel Conservation; Cost Index; 4D trajectories; Dijkstra’s algorithm; Waypoint navigation;

Base of Aircraft Data (BADA);

1. Introduction

Improving aircraft operational efficiency has become a dominant theme in air transportation, as the

airlines around the world have seen the price of fuel has risen sharply during the last decades. Currently,

aviation accounts for about 2% of total global CO2 emissions and about 12% of the CO2 from all

transportation source [1]. This increased fuel prices and environmental concerns have pushed airlines

to reduce fuel consumption and to find margins for performance improvements. Efforts to modernize

1

Annex B – Article

56

the aircraft fleet are limited by extremely slow and expensive process of new aircraft adoption, which

can take decades, therefore it is important to find different alternatives to reduce the fuel consumption

in current aircrafts, which will likely to share the sky with most modern aircraft in near future. One of

these alternatives is to optimize flight trajectories and traffic control procedure. The existing flight

planning techniques are suboptimal. Hence, a fuel optimal flight path can significantly save fuel.

A practical solution that reduces the cost associated with time and fuel consumption during flight is

the Cost Index (CI). The value of the CI reflects the relative effects of fuel cost on overall trip cost as

compared to time-related direct operating cost. For all the aircraft models, the minimum value of cost

index equal to zero results in maximum range airspeed and minimum trip fuel, but this configuration

ignores the time cost. If the cost index is maximum, the flight time is minimum, the velocity and the

Mach number are maximum, but ignores the fuel cost [2]. In this study the Cost Index assumes to be

zero as only fuel cost is taken into consideration.

~ (

~ €(

€ / )

/ )

hrTimeCostCI

FuelCost kg (1)

Recent studies propose that, during the takeoff and climb phase of the flight, accelerating and flap

retraction at a lower altitude than the typical 3000 ft decrease the fuel consumption, lower flap setting

cause low drag, resulting less fuel burn during climb, it also suggest that descending at a higher slope

angle than 30 enable the aircraft to save fuel [3], [4]. By improving the cruise speed and altitude profiles

is possible to reduce fuel burn in cruise phase, Patrick Hagelauer and Felix Mora camino conducted a

study based on a constant value of Cost Index for a given arrival time, in order to find the optimum

cruise speed and altitude profile [5].

Classical aircraft trajectory optimizations are solved by applying calculus of variations to determine

the optimality conditions, requiring the solution of non-linear Two-Point Boundary Value Problems

(TPBVPs) [6]. Alternatively, a more general solution to aircraft trajectory optimization can be obtained

by singular perturbation theory which approximates solutions of high order problems by the solution of

a series of lower order systems with the system dynamics separated into low and fast modes [7], [8].

With the tremendous advancement in numerical computing power, TPBVPs can be converted to

Annex B - Article

57

nonlinear programming problems that are solvable even for problems with many variables and

constraints using numerical algorithms such as direct collocation methods. Neglecting aircraft dynamics

and applying shortest path algorithms in graph theory, an optimal trajectory can be approximated by the

path that minimizes the total link cost connecting the origin and destination in a pre-defined network.

The graph methods often require large computation time and memory space but guarantee global

optimal solutions. In this paper the single source shortest path algorithm was used to generate the fuel

optimal trajectory.

This study is restricted to the climb, cruise and descent phases of the flight and ignores the takeoff

and landing approach, and assuming the initial and final waypoints are at altitude of 3000 feet, where

in the initial waypoint the aircraft begins the climb phase and in the final waypoint the aircraft begins

the landing approach. This work primarily attempts to quantify benefits of fuel optimal trajectory which

was found by implying the Dijkstra’s shortest path algorithm, a benefit is meant to imply a reduction in

fuel burn due to using the Dijkstra’s shortest path algorithm to the actual unimproved flight.

2. Problem Statement

The main goal of this paper is to find a fuel optimal path from a 4D waypoint networks. Most of the

approaches consider the waypoints defined by tridimensional coordinate positions ( , , )T

k k k kP h

k =1,2,…,N and do not take into account the time. By adding the arrival time restriction to the tri-

dimensional waypoint it is possible to define the 4D waypoints as ( , , , )T

k k k k kP h . Where:

, , ,k k k kh is respectively longitude, latitude, altitude and arrival time at waypoint kP .

The problem to be solved is to estimate associated travel time kd and consumed fuel kdf

between the waypoints in the 4D waypoint networks, then to use the value of associated consumed fuel

kdf between waypoints as edges and the waypoints as vertices in Dijkstra’s algorithm to find the fuel

optimal trajectory from initial waypoint to final waypoint from the 4D waypoint networks.

The following section propose a method that will determine the fuel optimal path along specified

Annex B – Article

58

waypoints from a 4D waypoint networks by implying the Dijkstra’s shortest path algorithm.

3. Proposed Method

To generate an fuel optimal trajectory from a set of waypoints in 4D waypoint networks requires

finding the associated fuel consumed kdf by the aircraft to go from one waypoint to the other, defined

as:

k nom kdf f d (2)

Where, nomf [kg/min] is the nominal fuel flow rate, kdf [kg] is the amount of fuel consumed by

the aircraft to go from waypoints 1kP to kP and, kd [min] is the amount of time needed to go

from waypoints 1kP to kP . Which can be described in the following equations:

1k k kdf f f (3)

1k k kd (4)

Where, kf [kg] and k [min] are respectively the fuel and time required to get to waypoint kP

from initial waypoint, 1kf [kg] and 1k [min] are respectively the fuel and time required to get to

waypoint 1kP from initial waypoint.

3.1 Dijkstra’s Algorithm

Dijkstra’s algorithm is a simple greedy algorithm to solve single source shortest paths problem. For

a given source vertex in the graph, the algorithm finds the path with lowest cost (i.e. the shortest path)

between that vertex and every other vertex [9] [10] [11]. It require three inputs (G, w, s) they are the

graph G, the weights w, and the source vertex s. Graphs are often used to model networks in which one

travels from one point to another. A graph G (V, E) refers to a collection of vertices V and a collection

Annex B - Article

59

of edges E that connect pairs of vertices and assuming all edges cost are non-negative, thus there are no

negative cycles and shortest paths exists for all vertices reachable from source vertex s. In this work the

waypoints of the 4D waypoint networks are the vertices, the initial waypoint is the source vertex, the

final waypoint is the destination vertex and the consumed fuel kdf by the aircraft between the pairs

of waypoints are the edges between these vertices (waypoints).

3.2 Modeling of 4D Waypoints Networks

Suppose the waypoints networks consists of N sets of waypoints, where 1P is the initial waypoint

and NP is the final waypoint. Each waypoint kP , (k =1,2,…., N) is defined by the geodetic coordinates

, ,k k kh , by considering the arrival time in each waypoint k , the waypoint kP can be described

as a four-dimensional state vector:

( , , , )T

k k k k kP h (5)

Where, k and k are the longitude and latitude of waypoint kP , kh is the altitude (with

respect to sea level). The following subsections represent the navigation model and constraints of 4D

waypoints networks.

3.2.1 Navigation Model

The following differential equations model the dynamics of the navigation process:

cos sin

( )cose

V

R h

(6)

cos cos

( )e

V

R h

(7)

sinh V (8)

Annex B – Article

60

1V u (9)

2u (10)

3u (11)

Where, V = flight velocity, = flight path angle, =heading (with respect to the geographical

north), = longitude, = latitude, h = the altitude (with respect to sea level), eR is the Earth

radius. The variables 1u , 2u , and 3u are respectively the acceleration, the flight path angle rate

and the heading rate. The state vector x and control vector u of the above model are described

respectively as:

( , , , , , )Tx h V (12)

1 2 3,( ), Tu u u u (13)

3.2.2 Navigation Constraints

The real world flight operate under several constraints, due to aircraft performance, aerodynamic

structural limits, safety reasons, the mission and other factor.

min max a a a (14)

min max (15)

min max V V V (16)

3.3 Arrival Time of Each Waypoint

The 4D navigation consists of traveling through a sequence of predefined waypoints in a given time

Annex B - Article

61

of flight, which in turn define the trajectory of the flight. Assuming that the waypoint kP is already

defined by the geodetic coordinates longitude k , latitude k , and altitude kh , where the scheduled

time of arrival k at this waypoint is unknown. To compute this unknown arrival time of each

waypoint, the distance between this waypoint and its previous waypoint from where the aircraft is

arriving and the average velocity of the aircraft between this two waypoints is required.

The trajectory generation requires a geocentric coordinates system. To calculate the distance between

two waypoints, the 3D waypoint need to be transformed from usual geodetic coordinates system to

geocentric coordinates system. The 3D waypoint kP is defined by the following way:

( , , )T

k k k kP h (17)

Now to transform these geodetic coordinates to geocentric coordinates, the following equations are

required [12].

( )cos cosk k k k kX N h (18)

( )cos sink k k k kY N h (19)

2[ (1 ) ]sink k k kZ N e h (20)

Being a is the Earth semi major axis and e its eccentricity, kN can be calculated as follows:

2 2

a

1 sink

k

Ne

(21)

After transforming the 3D waypoint from geodetic coordinates to geocentric coordinates system,

Now it is possible to calculate kd (the distance between two waypoints 1kP to kP ) by using the

following equation:

2 2 2

1 1 1(X ) (Y ) (Z )k k k k k k kd X Y Z (22)

To estimate the appropriate velocity of the aircraft kV at any waypoint kP , the following equation

can be used:

Annex B – Article

62

2k

k L

WV

C S (23)

Where, W is the aircraft nominal weight, LC is the lift coefficient, S is the wing area of the

aircraft and k is the air density (varies with altitude) at waypoint kP . It is possible to get the

appropriate velocity at any waypoint kP from Base of Aircraft Data (BADA), where true air speed,

VTAS [kt] is specified for different aircraft for different flight level and phases of the flights [13].

By using the distance between two waypoints and the velocity of the aircraft in both waypoints the

arrival time of each waypoint can be computed as follow:

1

2

kk

k k

dd

V V

(24)

Where, kd is the time needed to go from waypoint 1kP to kP . So the arrival time of waypoint

kP from the initial waypoint can be described as follow:

1k k kd (25)

In practice, the aircraft may not pass through the waypoint kP exactly at this specified time k

due to disturbances. Therefore, an appropriate way is rather imposing a time tolerance in (equation

25).

1k k kd (26)

Where, is the tolerance time interval for arrival at a determined waypoint [1.0 ≤ ϵ ≤ 1.4], if

the altitude of waypoints 1kP and kP are same, then the tolerance ϵ can be assumed 1.

Annex B - Article

63

3.4 Fuel Consumption Model

This Subsection develops the fuel consumption model for commercial flights. In commercial flight

the rate of fuel burn depends on ambient temperature, aircraft speed, and aircraft altitude. It also depend

on the aircraft weight which changes as fuel burned. The Base of Aircraft Data (BADA) model provides

coefficients that allows to calculate the thrust specific fuel consumption and different thrust level

Thr , which can be used to calculate the nominal fuel flow rates nomf in different phase of the flight

[14].

3.4.1 Nominal Fuel Flow Rate

The nominal fuel flow rate, nomf [kg/min], can be calculated using the thrust and thrust specific fuel

consumption as follows:

nomf Thr (27)

The thrust varies with different flight phases, thus the thrust in (equation 27) depends on the phase

which the aircraft is flying, i.e. if the aircraft is flying in climb, cruise or descent phase the thrust of this

equation will be respectively climb, cruise or descent thrust. The thrust specific fuel consumption

and different thrust level Thr are explained in the following subsections. This expression can be used

in all flight phases except during idle descent and cruise, where, the following expressions are to be

used.

The minimum fuel flow rate, minf [kg/min], corresponding to idle thrust descent conditions, is

specified as a function of the geo-potential pressure altitude, pH [ft], that is:

3

4

min (1 )p

f

f

Hf C

C (28)

Where, 3f

C and 4f

C are the descent fuel flow coefficients specified in BADA tables. The idle

thrust part of the descent stops when the aircraft switches to approach and landing configuration, at this

Annex B – Article

64

point thrust is generally increased. Hence, the calculation of the fuel flow during approach and landing

phases shall be based on the nominal fuel flow rate, and limited to the minimum fuel flow:

/ minMAX( , )ap ld nomf f f (29)

To calculate the cruise fuel flow rate, crf [kg/min], a cruise fuel flow correction coefficient, crfC is

needed with the thrust specific fuel consumption, [kg/ (min.KN)] and the thrust, Thr [N],which can

be expressed in the following form:

crcr ff Thr C (30)

For the moment the cruise fuel flow correction factor has been established for a number of aircraft

types whenever the reference data for cruise fuel consumption is available. This factor has been set to

1 for all the other aircraft models.

For now the nominal fuel flow rate nomf [kg/min], (equation27) can be used for different phase of

the flight, as the cruise fuel flow correction factor crfC has been set to 1 for most of the aircraft models,

and if the thrust is not ideal during descent the fuel flow rate is based on the nominal fuel flow rate.

3.4.2 Thrust Specific Fuel Consumption

For jet engine the thrust specific fuel consumption, [kg/ (min.KN)], is an engineering term that is

used to describe the fuel efficiency of an engine design with respect to thrust output, and is specified as

a function of the true airspeed, TASV [kt]:

1

2

(1 )TASf

f

VC

C (31)

Where,1f

C and 2f

C are the thrust specific fuel consumption coefficients specified in BADA tables.

3.4.3 Engine Thrust

To calculate the nominal fuel flow rate, it is necessary to calculate the engine thrust at different phase

of flight. The BADA model provides coefficients that allow the calculation of the following thrust levels:

Annex B - Article

65

3.4.3.1 Maximum Climb and Take-off Thrust

The maximum climb thrust at standard atmosphere conditions, maxclimb( )ISAThr , is calculated

Newton’s as a function of geo potential pressure altitude, pH [ft]; true airspeed, TASV [kt]; and

temperature deviation from standard atmosphere, T [K]. Here only jet engine type was considered,

The equation for maximum climb and take-off thrust as follows:

,1 ,3

,2

2

maxclimb( )ISA= (1 )c c

c

p

T T p

T

HThr C C H

C (32)

Where, ,1cTC ,

,2cTC and ,3cTC are climb thrust coefficient specified in the BADA tables.

The maximum climb thrust is corrected for temperature deviations from standard atmosphere, T ,

in the following manner:

,5maxclimb maxclimb( )ISA (1 )cT effThr Thr C T (33)

Where,

,4ceff TT T C (34)

With the limits,

,50.0 0.4

cT effC T (35)

,50.0

cTC (36)

Where, ,4cTC and

,5cTC are thrust temperature coefficient specified in the BADA tables. This

maximum climb thrust is used for both take-off and climb phases.

3.4.3.2 Maximum Cruise Thrust

The normal cruise thrust is by definition set equal to drag ( Thr D ). However, the maximum

amount of thrust available in cruise situation is limited. The maximum cruise thrust is calculated as a

ratio of the maximum climb thrust as follows:

cruise maxclimb( )MAXcrTThr C Thr (37)

The Maximum cruise thrust coefficient crTC is currently uniformly set for all aircraft of value 0.95.

Annex B – Article

66

3.4.3.3 Descent Thrust

Descent thrust is calculated as a ratio of the maximum climb thrust, with different correction factors

used for high and low altitudes, and approach and landing configurations, that is:

If ,p p desH H

des,highdes,high maxclimbTThr C Thr (38)

Where, ,p desH is the transition altitude, des,highTC is high altitude descent thrust coefficient specified

in BADA tables.

If ,p p desH H

Cruise configuration: des,lowdes,low maxclimbTThr C Thr (39)

Where, des,lowTC is the low altitude descent thrust coefficient specified in BADA tables.

Approach configuration: des,appdes,app maxclimbTThr C Thr (40)

Where, des,appTC is the approach thrust coefficient specified in BADA tables.

Landing Configuration: des,lddes,ld maxclimbTThr C Thr (41)

Where, des,ldTC is the landing thrust coefficient specified in BADA tables.

4. Simulation and Result

In this section the simulation and result of the fuel optimal trajectories are shown for three different

length (short, medium and long haul) trajectories considering no wind. All the analysis of the simulation

has been done using Matlab 2013a. The coordinates of the trajectory were chosen using skyvector

website [15] .

Annex B - Article

67

4.1 Short Haul Flight

To analyze the short haul flight, the flight from Lisbon to Geneva was considered. The 4D waypoint

network of this short haul flight consists of two trajectories, and has total 22 waypoints including the

initial and final waypoints, and each trajectory has 12 waypoints including the initial and final

waypoints. Airplane A1 (which is a short medium range twinjet narrow body airliner) was used to

analyze the flight trajectories. (table 1 and 2) Show the waypoints lists for both of the trajectories. Each

waypoint is defined in geodetic coordinates ( , , h ) and their associated distance kd , travel time

kd and consumed fuel kdf between the waypoints are also shown.

Table 1: List of waypoints in 1st trajectory for short haul flight

waypoint

[deg]

[deg]

h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -8.9083 39.1087 10000 9.249827 2.371751 291.0612

P3 -8.624 39.33417 20000 19.01912 3.437191 348.0156

P4 -7.7987 39.8783 33000 50.39803 7.295251 535.3803

P5 -6.9993 40.2513 39000 43.16538 5.748997 294.6361

P6 -3.3707 43.227 39000 242.0397 32.48854 1176.085

P7 0.1303 44.729 39000 176.7117 23.71969 858.6528

P8 3.5963 44.9543 39000 148.8367 19.97808 723.2065

P9 3.87217 45.1205 33000 15.45767 2.058735 7.823191

P10 4.6985 45.41983 20000 39.46054 5.71202 32.05871

P11 5.336 45.606 10000 29.19265 5.275779 42.20624

Final (P22) 5.7553 45.884 3000 24.30017 5.618537 54.83692

Total 797.8314 113.7046 4363.963

The trajectories were chosen such a way that the climb and descent phases of the first trajectory are

smaller than the second trajectory but the cruise phase of the first trajectory is bigger than the second

trajectory, therefore the total distance from the initial waypoint to final waypoint for both of the

trajectories are more or less same. In order to find the fuel optimal trajectory from the 4D waypoint

network possible connection between waypoints in both trajectories were established, and their

Annex B – Article

68

associated distance kd , travel time kd and consumed fuel kdf between these possible

waypoints connections were calculated.

Table 2: List of waypoints in 2nd trajectory for short haul flight.

waypoint

[deg]

[deg]

h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P12 -8.835 39.0883 10000 11.16491 2.862798 351.3225

P13 -8.49983 39.30183 20000 20.28586 3.66612 371.1947

P14 -7.59217 39.7763 33000 50.94383 7.374258 541.1783

P15 -6.715 40.11317 39000 45.32355 6.036433 309.3672

P16 -1.765 41.631 39000 243.352 32.6647 1182.462

P17 0.5565 43.9993 39000 175.4797 23.55432 852.6664

P18 3.6277 44.85317 39000 141.8812 19.04446 689.4095

P19 3.9993 44.9983 33000 18.13834 2.415762 9.179894

P20 4.8617 45.261 20000 39.98448 5.787862 32.48437

P21 5.4975 45.5203 10000 31.13026 5.625951 45.00761

Final (P22) 5.7553 45.884 3000 24.40439 5.642634 55.0721

Total 802.0885 114.6753 4439.345

The fuel optimal trajectory was generated from the 4D waypoint network using the Dijkstra’s shortest

path algorithm. The fuel optimal trajectory contains 9 waypoints [initial waypoint (P1)→ P2→ P3→

P4→ P5→ P18→ P19→ P11→ final waypoint (P22)], the distance between the initial and final waypoints

in fuel optimal trajectory is 777.7 nm. The comparison of fuel consumed in different phases of flight

for these two trajectories and fuel optimal trajectory are shown in (table 3).

Table 3: Fuel consumed from initial to final waypoint in different trajectories for short haul flight.



1 1469.1 2757.9 136.9 4363.9

2 1573.1 2724.5 141.7 4439.3

Fuel optimal 1469.1 2652.8 136.1 4258

As seen in (table 3), by using the fuel optimal trajectory for the short haul flight (Lisbon – Geneva)

the aircraft consumes 105.9 kg of less fuel than the first trajectory, which is equivalent to 2.4% less

fuel than the first trajectory and consumes 181.3 kg of less fuel than the second trajectory, which is

equivalent to 4.1% less fuel than the second trajectory. The fuel optimal trajectory in 3D is shown in

Annex B - Article

69

(figure 1).

Figure 1: 3D fuel optimal trajectory in geocentric coordinates for short haul flight.

In (figure 1) the fuel optimal trajectory is represented by the blue line and the red circles around the

trajectory are the waypoints.

4.2 Medium Haul Flight

To analyze the medium haul flight, the flight from Lisbon to Stockholm was considered. There are

also two trajectories between the initial and final waypoints in the 4D waypoint network, each

trajectories has 13 waypoints including the initial and final waypoints, and total 24 waypoints are there

in the 4D waypoint network including the initial and final waypoint. Airplane A2 (which is a long range

wide body twinjet airliner) was used to analyze the flight trajectories. (table 4 and 5) Show the

waypoints lists for both of the trajectories.

Table 4: List of waypoints in 1st trajectory for the medium haul flight

waypoint

[deg]

[deg]

h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -8.9373 39.1525 10000 10.63985 2.665515 1070.631

P3 -8.643 39.6385 24000 32.29915 5.529098 1724.94

P4 -8.0007 40.5407 37000 61.77433 8.291856 1859.863

P5 -7.7847 40.942 41000 26.05963 3.243938 589.0991

Annex B – Article

70

waypoint

[deg]

[deg]

h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

P6 -4.752 46.1707 41000 340.8976 42.43539 5385.051

P7 1.07817 49.6497 41000 314.9914 39.21055 4975.819

P8 9.07617 53.515 41000 379.0242 47.18143 5987.324

P9 14.4575 57.839 41000 318.0823 39.59531 5024.645

P10 14.75417 58.0005 37000 13.62262 1.695762 31.54117

P11 15.78983 58.624 24000 49.92285 6.807662 161.8522

P12 17.0113 59.2063 10000 51.77227 9.043191 287.5735

Final (P24) 17.73983 59.3445 3000 23.95284 5.538229 212.0034

Total 1623.039 211.2379 27310.34

Table 5: List of waypoints in 2nd trajectory for the medium haul flight

waypoint

[deg]

[deg]

h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P13 -8.8405 39.1047 10000 11.47263 2.874146 1154.43

P14 -8.455 39.576 24000 33.57806 5.748028 1793.241

P15 -7.808 40.4545 37000 60.65319 8.141368 1826.109

P16 -7.413 40.8377 41000 29.26358 3.642769 661.5269

P17 -0.2773 44.76 41000 393.6387 49.00066 6218.184

P18 4.6187 50.023 41000 374.0856 46.56667 5909.311

P19 10.92483 54.85883 41000 371.7582 46.27695 5872.545

P20 15.0405 57.2845 41000 201.3458 25.06379 3180.595

P21 15.39 57.4757 37000 16.19891 2.016462 37.50619

P22 16.496 58.17 24000 54.91818 7.488843 178.0473

P23 17.504 58.94117 10000 56.26139 9.827318 312.5087

Final (P24) 17.73983 59.3445 3000 25.36559 5.864875 224.5074

Total 1628.54 212.5119 27368.51

The fuel optimal trajectory contains 9 waypoints [initial waypoint (P1)→ P2→ P3→ P4→ P5→ P20→

P21→ P23→ final waypoint (P24)], which was generated implying Dijkstra’s algorithm, the distance

between the initial and final waypoints in fuel optimal trajectory for this flight is 1596.5 nm.

Table 6: Fuel consumed from initial to final waypoint in different trajectories for medium haul

flight



1 5244.5 21372.8 692.97 27310.3

2 5435.3 21180.6 752.6 27368.5

Fuel optimal 5244.5 20744.4 742.2 26731.1

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The comparison of consumed fuel in different phases of flight for different trajectories including the

fuel optimal trajectory are shown in (table 6).

From the initial waypoint to reach the final waypoint using the fuel optimal trajectory the aircraft

consumes 579.2 kg of less fuel than the first trajectory and consumes 637.4 kg of less fuel than the

second trajectory. In other word by using the fuel optimal trajectory for the medium haul flight (Lisbon

– Stockholm) the aircraft consumes 2.1% less fuel than the first trajectory, and 2.3% less fuel than the

second trajectory. The fuel optimal trajectory in three dimension is shown in (figure 2).

Figure 2: 3D fuel optimal trajectory in geocentric coordinates for medium haul flight.

The blue curve in (figure 2) corresponds to the fuel optimal trajectory for the medium haul flight

(Lisbon – Stockholm) and the red circles around the curve are the waypoints of the fuel optimal

trajectory.

4.3 Long Haul Flight

The flight from Lisbon to Montreal was considered to analyze the long haul flight. The 4D waypoint

network of the long haul flight also consists of two trajectories between the initial and final waypoints,

each trajectories has 14 waypoints including the initial and final waypoints, and total 26 waypoints are

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in the whole 4D waypoint network including initial and final waypoints. Airplane A2 (which is a long

range wide body twinjet airliner) was used to analyze the flight trajectories. The waypoints lists of both

trajectories are shown in (table 7 and 8).

Table 7: List of waypoints in 1st trajectory of long haul flight

waypoint

[deg]

[deg]

h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P2 -9.22083 39.1 10000 10.5683 2.64759 1063.431

P3 -9.8425 39.3857 24000 33.76592 5.780186 1803.274

P4 -10.9352 39.9543 37000 61.15747 8.209056 1841.291

P5 -11.74 40.32817 43000 43.3812 5.40015 959.6066

P6 -22.5022 43.5023 43000 518.4016 64.53132 8227.743

P7 -35.132 44.73617 43000 550.8432 68.56969 8742.636

P8 -44.9847 47.4295 43000 442.3112 55.05948 7020.084

P9 -58.667 48.5647 43000 555.3268 69.12781 8813.796

P10 -70.3565 46.4947 43000 491.3069 61.15854 7797.714

P11 -70.809 46.4135 37000 19.45513 2.421801 43.59241

P12 -71.9328 46.21 24000 48.42787 6.6038 157.0053

P13 -73.0908 45.9335 10000 51.23244 8.948898 284.575

Final (P26) -73.6412 45.7947 3000 24.56775 5.680405 217.4459

Total 2850.746 364.1387 46972.19

Table 8: List of waypoints in 2nd trajectory of long haul flight

waypoint

[deg]

[deg]

h

[feet] kd

[nm]

kd

[min]

kdf

[kg]

Initial (P1) -9.0405 38.9955 3000 0 0 0

P14 -9.2387 39.05683 10000 10.03791 2.514717 1010.061

P15 -9.94583 39.2603 24000 35.28932 6.040968 1884.631

P16 -11.2588 39.4515 37000 62.29715 8.362034 1875.604

P17 -11.9673 39.9493 43000 44.44929 5.533107 983.2331

P18 -24.8818 42.095 43000 600.2876 74.7246 9527.386

P19 -34.374 43.996 43000 433.2285 53.92886 6875.93

P20 -44.3497 46.44083 43000 448.0273 55.77103 7110.806

P21 -61.77 47.44 43000 717.2308 89.28184 11383.43

P22 -70.163 46.2543 43000 353.4368 43.99628 5609.525

P23 -70.557 46.102 37000 18.61899 2.317716 41.71889

P24 -71.739 45.9043 24000 51.21564 6.98395 166.0434

P25 -72.9952 45.81 10000 53.07175 9.270174 294.7915

Final (P26) -73.6412 45.7947 3000 27.16449 6.280807 240.4293

Total 2854.355 365.0061 47003.6

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Possible connection between waypoints in both trajectories were established, and their associated

distance kd , travel time kd and consumed fuel kdf between these possible waypoints

connections were calculated in order to find the fuel optimal trajectory by implying Dijkstra’s shortest

path algorithm.

The fuel optimal trajectory of the flight between Lisbon to Montreal contains 9 waypoints [initial

waypoint (P1)→ P14→ P3→ P4→ P5→ P22→ P11→ P13→ final waypoint (P26)], the distance between

the initial and final waypoints in fuel optimal trajectory is 2777 nm. The comparison of fuel consumed

in different phases of flight for different trajectories and the fuel optimal trajectory are shown in (table

9).

Table 9: Fuel consumed from initial to final waypoint in different trajectories for long haul flight



1 5667.6 40601.9 702.6 46972.2

2 5753.5 40507.1 742.98 47003.6

Fuel optimal 5652.1 39284.9 712.5 45649.5

Using the fuel optimal trajectory in long haul flight the aircraft consumed 1322.7 kg of less fuel than

the first trajectory and consumed 1354.1 kg less fuel than the second trajectory. So in the fuel optimal

trajectory for the long haul flight the aircraft consumes 2.8% less fuel than the first trajectory, and 2.9%

less fuel than the second trajectory. The 3D fuel optimal trajectory is shown in (figure 3).

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Figure 3: 3D fuel optimal trajectory in geocentric coordinates for long haul flight.

The blue curve in (figure 3) is the fuel optimal trajectory and the red circles around it are the

waypoints of that trajectory. In this long haul flight the cruise phase is really large compare to its climb

and descent phases, therefore the waypoints in the climb and descent phases are seems really close to

each other in the figure.

5. Conclusion

This study is based on finding the fuel optimal trajectories of climb, cruise and descent phases of the

flight, without the takeoff and landing phases of the flight. In this work several steps were made in order

to achieve a complete trajectory from a 4D waypoint networks that optimize the fuel consumption. This

study uses Dijkstra’s shortest path algorithm that find an fuel optimal trajectory from a given 4D

waypoint networks, this technique was used to compare different length (short, medium and long haul)

flights.

The analysis results show promising potential for reduction of consumed fuel in different flights via

using the Dijkstra’s shortest path algorithm, across a range of common aircrafts and routes. The results

suggest that by flying fuel optimal trajectory for short haul flight, it is possible to save 2.4−4.1% on

fuel burn, which is equivalent to 105.9 – 181.3 kilograms of fuel for Airplane A1. In medium haul flight

by flying the fuel optimal trajectory can potentially save 2.1−2.3% fuel, reducing fuel burn by 579.2 –

637.4 kilograms for Airplane A2. For long haul flight it is possible to save 2.8−2.9% on fuel burn,

which is equivalent to 1322.7 – 1354.1 kilogram of fuel for Airplane A2. In general the savings of the

fuel is proportional to the trip lengths, and depends on the aircraft types.

6. References

[1] ICAO, "ICAO Environmental Report: Aviation and Climate change," ICAO, 2013.

[2] B. Robenson, "Fuel Conservation Strategies: Cost Index Explained," Boeing, 2007, pp.26-28.

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[3] W. Roberson and J. A. Johns, "Fuel Conservation strategies: Takeoff and Climb," Boeing, 2008,

pp. 25-28.

[4] W. Roberson and J. A. Johns, "Fuel Conservation Strategies: Descent and Approach," Boeing,

2008, pp. 25-28.

[5] Hagelauer, P. and Mora-Camino, F., "Evaluation of Practical Solutions for Onboard Aircraft Four-

Dimensional Guidance," Journal of Guidance, Control, and Dynamics, Vol. 20, No. 5, 1997, pp.

1052 - 1054.

[6] Bryson, A. E. and Y. C. Ho, "Applied Optimal Control: Optimization, Estimation and, Control",

Taylor & Francis, New York, 1975, Ch. 2, pp. 42-89.

[7] Calise, A. J. "Extended Energy Management for Flight Performance Optimization, AIAA Journal,

VOl. 15, No. 3, March 1977, pp. 314-321.

[8] Calise A. J. and D. D. Moerder, "Singular Perturbation Techniques for Real Time Aircraft

Trajectory Optimization and Control", NASA CR-3597, 1982.

[9] Cormen T. H., Leiserson C. E., Rivest R. L., and Stein C., "Introduction to Algorithms", London,

England: The MIT Press, 2009 3rd ed, pp. 658-664.

[10] Hart C., "Graph Theory Topics in Computer Networking", 2013, pp. 13-20.

[11] Dasgupta S., Papadimitriou C. H. and Vazirani U. V.," Algorithms", McGraw-Hill, New York, July

18, 2006, pp. 112-118.

[12] Seemkooei A. A., "Comparison of different algorithm to transform geocentric to geodetic

coordinates," Survey Review 36, 286 October 2002. pp. 627-632.

[13] "Aircraft Performance Summary Tables for the Base of Aircraft Data (BADA)," Eurocontrol

Experimental Centre, Revision 3.4, June 2002.

[14] "User Manual for the Base of Aircraft Data (BADA)," Eurocontrol Experimental Centre, Revision

3.9, April 2011.

[15] "SkyVector," [Online]. Available: https://skyvector.com/. [Accessed October 2015].

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